Heronian triangles

Heronian triangles
You are encouraged to solve this task according to the task description, using any language you may know.

Hero's formula for the area of a triangle given the length of its three sides   a,   b,   and   c   is given by:

${\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}},}$

where   s   is half the perimeter of the triangle; that is,

${\displaystyle s={\frac {a+b+c}{2}}.}$

Heronian triangles are triangles whose sides and area are all integers.

An example is the triangle with sides   3, 4, 5   whose area is   6   (and whose perimeter is   12).

Note that any triangle whose sides are all an integer multiple of   3, 4, 5;   such as   6, 8, 10,   will also be a Heronian triangle.

Define a Primitive Heronian triangle as a Heronian triangle where the greatest common divisor of all three sides is   1   (unity).

This will exclude, for example, triangle   6, 8, 10.

1. Create a named function/method/procedure/... that implements Hero's formula.
2. Use the function to generate all the primitive Heronian triangles with sides <= 200.
3. Show the count of how many triangles are found.
4. Order the triangles by first increasing area, then by increasing perimeter, then by increasing maximum side lengths
5. Show the first ten ordered triangles in a table of sides, perimeter, and area.
6. Show a similar ordered table for those triangles with area = 210

Show all output here.

Note: when generating triangles it may help to restrict ${\displaystyle a<=b<=c}$

11l

Translation of: Python
F gcd(=u, =v)
L v != 0
(u, v) = (v, u % v)
R abs(u)

F hero(a, b, c)
V s = (a + b + c) / 2
V a2 = s * (s - a) * (s - b) * (s - c)
R I a2 > 0 {sqrt(a2)} E 0

F is_heronian(a, b, c)
V x = hero(a, b, c)
R x > 0 & fract(x) == 0

F gcd3(x, y, z)
R gcd(gcd(x, y), z)

V MAXSIDE = 200
[(Int, Int, Int)] h
L(x) 1..MAXSIDE
L(y) x..MAXSIDE
L(z) y..MAXSIDE
I (x + y > z) & gcd3(x, y, z) == 1 & is_heronian(x, y, z)
h [+]= (x, y, z)

h = sorted(h, key' x -> (hero(x[0], x[1], x[2]), sum(x), (x[2], x[1], x[0])))

print(‘Primitive Heronian triangles with sides up to #.: #.’.format(MAXSIDE, h.len))
print("\nFirst ten when ordered by increasing area, then perimeter, then maximum sides:")
print(h[0.<10].map3((x, y, z) -> ‘  #14 perim: #3 area: #.’.format(String((x, y, z)), x + y + z, hero(x, y, z))).join("\n"))
print("\nAll with area 210 subject to the previous ordering:")
print(h.filter3((x, y, z) -> hero(x, y, z) == 210).map3((x, y, z) -> ‘  #14 perim: #3 area: #.’.format(String((x, y, z)), x + y + z, hero(x, y, z))).join("\n"))
Output:
Primitive Heronian triangles with sides up to 200: 517

First ten when ordered by increasing area, then perimeter, then maximum sides:
(3, 4, 5) perim:  12 area: 6
(5, 5, 6) perim:  16 area: 12
(5, 5, 8) perim:  18 area: 12
(4, 13, 15) perim:  32 area: 24
(5, 12, 13) perim:  30 area: 30
(9, 10, 17) perim:  36 area: 36
(3, 25, 26) perim:  54 area: 36
(7, 15, 20) perim:  42 area: 42
(10, 13, 13) perim:  36 area: 60
(8, 15, 17) perim:  40 area: 60

All with area 210 subject to the previous ordering:
(17, 25, 28) perim:  70 area: 210
(20, 21, 29) perim:  70 area: 210
(12, 35, 37) perim:  84 area: 210
(17, 28, 39) perim:  84 area: 210
(7, 65, 68) perim: 140 area: 210
(3, 148, 149) perim: 300 area: 210


with Ada.Containers.Indefinite_Ordered_Sets;
procedure Heronian is
use Int_IO;

-- ----- Some math...
function GCD (A, B : in Natural) return Natural is (if B = 0 then A else GCD (B, A mod B));

function Int_Sqrt (N : in Natural) return Natural is
R1 : Natural := N;
R2 : Natural;
begin
if N <= 1 then
return N;
end if;
loop
R2 := (R1+N/R1)/2;
if R2 >= R1 then
return R1;
end if;
R1 := R2;
end loop;
end Int_Sqrt;

-- ----- Defines the triangle with sides as discriminants and a constructor which will
-- compute its other characteristics
type t_Triangle (A, B, C : Positive) is new Ada.Finalization.Controlled with record
Is_Heronian  : Boolean;
Perimeter    : Positive;
Area         : Natural;
end record;

overriding procedure Initialize (Self : in out t_Triangle) is
-- Let's stick to integer computations, therefore a modified hero's formula
-- will be used : S*(S-a)*(S-b)*(S-c) = (a+b+c)*(-a+b+c)*(a-b+c)*(a+b-c)/16
-- This will require long integers because at max side size, the product
-- before /16 excesses 2^31
Long_Product  : Long_Long_Integer;
Short_Product : Natural;
begin
Self.Perimeter   := Self.A + Self.B + Self.C;
Long_Product     :=   Long_Long_Integer(Self.Perimeter)
* Long_Long_Integer(- Self.A + Self.B + Self.C)
* Long_Long_Integer(  Self.A - Self.B + Self.C)
* Long_Long_Integer(  Self.A + Self.B - Self.C);
Short_Product    := Natural(Long_Product / 16);
Self.Area        := Int_Sqrt (Short_Product);
Self.Is_Heronian := (Long_Product mod 16 = 0) and (Self.Area * Self.Area = Short_Product);
end Initialize;

-- ----- Ordering triangles with criteria (Area,Perimeter,A,B,C)
function "<" (Left, Right : in t_Triangle) return Boolean is
(Left.Area      < Right.Area      or else (Left.Area      = Right.Area      and then
(Left.Perimeter < Right.Perimeter or else (Left.Perimeter = Right.Perimeter and then
(Left.A         < Right.A         or else (Left.A         = Right.A         and then
(Left.B         < Right.B         or else (Left.B         = Right.B         and then
Left.C         < Right.C))))))));
package Triangle_Lists is new Ada.Containers.Indefinite_Ordered_Sets (t_Triangle);
use Triangle_Lists;

-- ----- Displaying triangle characteristics
Header : constant String := "  A   B   C Per  Area" & ASCII.LF & "---+---+---+---+-----";
procedure Put_Triangle (Position : Cursor) is
Triangle : constant t_Triangle := Element(Position);
begin
Put(Triangle.A, 3);
Put(Triangle.B, 4);
Put(Triangle.C, 4);
Put(Triangle.Perimeter, 4);
Put(Triangle.Area,      6);
New_Line;
end Put_Triangle;

-- ----- Global variables
Triangles : Set := Empty_Set;
-- Instead of constructing two sets, or browsing all the beginning of the set during
-- the second output, start/end cursors will be updated during the insertions.
First_201 : Cursor := No_Element;
Last_201  : Cursor := No_Element;

procedure Memorize_Triangle (A, B, C : in Positive) is
Candidate : t_Triangle(A, B, C);
Position  : Cursor;
Dummy     : Boolean;
begin
if Candidate.Is_Heronian then
Triangles.Insert (Candidate, Position, Dummy);
if Candidate.Area = 210 then
First_201 := (if    First_201 = No_Element then Position
elsif Position < First_201   then Position
else  First_201);
Last_201 :=  (if    Last_201  = No_Element then Position
elsif Last_201  < Position   then Position
else  Last_201);
end if;
end if;
end Memorize_Triangle;

begin
-- Loops restrict to unique A,B,C (ensured by A <= B <= C) with sides < 200 and for
-- which a triangle is constructible : C is not greater than B+A (flat triangle)
for A in 1..200 loop
for B in A..200 loop
for C in B..Integer'Min(A+B-1,200) loop
-- Filter non-primitive triangles
if GCD(GCD(A,B),C) = 1 then
Memorize_Triangle (A, B, C);
end if;
end loop;
end loop;
end loop;

Put_Line (Triangles.Length'Img & " heronian triangles found :");
Triangles.Iterate (Process => Put_Triangle'Access);
New_Line;

Put_Line ("Heronian triangles with area = 201");
declare
Position : Cursor := First_201;
begin
loop
Put_Triangle (Position);
exit when Position = Last_201;
Position := Next(Position);
end loop;
end;
end Heronian;

Output:
 517 heronian triangles found :
A   B   C Per  Area
---+---+---+---+-----
3   4   5  12     6
5   5   6  16    12
5   5   8  18    12
4  13  15  32    24
5  12  13  30    30
9  10  17  36    36
3  25  26  54    36
7  15  20  42    42
10  13  13  36    60
8  15  17  40    60

...

Heronian triangles with area = 201
A   B   C Per  Area
---+---+---+---+-----
17  25  28  70   210
20  21  29  70   210
12  35  37  84   210
17  28  39  84   210
7  65  68 140   210
3 148 149 300   210

ALGOL 68

Translation of: Lua
# mode to hold details of a Heronian triangle #
MODE HERONIAN = STRUCT( INT a, b, c, area, perimeter );
# returns the details of the Heronian Triangle with sides a, b, c or nil if it isn't one #
PROC try ht = ( INT a, b, c )REF HERONIAN:
BEGIN
REF HERONIAN    t := NIL;
REAL            s  = ( a + b + c ) / 2;
REAL area squared  = s * ( s - a ) * ( s - b ) * ( s - c );
IF area squared > 0 THEN
# a, b, c does form a triangle #
REAL area      = sqrt( area squared );
IF ENTIER area = area THEN
# the area is integral so the triangle is Heronian #
t := HEAP HERONIAN := ( a, b, c, ENTIER area, a + b + c )
FI
FI;
t
END # try ht # ;
# returns the GCD of a and b #
PROC gcd = ( INT a, b )INT: IF b = 0 THEN a ELSE gcd( b, a MOD b ) FI;
# prints the details of the Heronian triangle t #
PROC ht print = ( REF HERONIAN t )VOID:
print( ( whole( a OF t, -4 ), whole( b OF t, -5 ), whole( c OF t, -5 ), whole( area OF t, -5 ), whole( perimeter OF t, -10 ), newline ) );
# prints headings for the Heronian Triangle table #
PROC ht title = VOID: print( ( "   a    b    c area perimeter", newline, "---- ---- ---- ---- ---------", newline ) );

BEGIN
# construct ht as a table of the Heronian Triangles with sides up to 200 #
[ 1 : 1000 ]REF HERONIAN ht;
REF HERONIAN             t;
INT                      ht count := 0;

FOR c TO 200 DO
FOR b TO c DO
FOR a TO b DO
IF gcd( gcd( a, b ), c ) = 1 THEN
t := try ht( a, b, c );
IF REF HERONIAN(t) ISNT REF HERONIAN(NIL) THEN
ht[ ht count +:= 1 ] := t
FI
FI
OD
OD
OD;

# sort the table on ascending area, perimeter and max side length #
# note we constructed the triangles with c as the longest side    #
BEGIN
INT          lower := 1, upper := ht count;
WHILE upper        := upper - 1;
BOOL swapped := FALSE;
FOR i FROM lower TO upper DO
REF HERONIAN h := ht[ i     ];
REF HERONIAN k := ht[ i + 1 ];
IF area OF k < area OF h OR (   area OF k =  area OF h
AND (  perimeter OF k <  perimeter OF h
OR (   perimeter OF k = perimeter OF h
AND c OF k < c OF h
)
)
)
THEN
ht[ i     ] := k;
ht[ i + 1 ] := h;
swapped     := TRUE
FI
OD;
swapped
DO SKIP OD;

# display the triangles #
print( ( "There are ", whole( ht count, 0 ), " Heronian triangles with sides up to 200", newline ) );
ht title;
FOR ht pos TO 10 DO ht print( ht( ht pos ) ) OD;
print( ( " ...", newline ) );
print( ( "Heronian triangles with area 210:", newline ) );
ht title;
FOR ht pos TO ht count DO
REF HERONIAN t := ht[ ht pos ];
IF area OF t = 210 THEN ht print( t ) FI
OD
END
END
Output:
There are 517 Heronian triangles with sides up to 200
a    b    c area perimeter
---- ---- ---- ---- ---------
3    4    5    6        12
5    5    6   12        16
5    5    8   12        18
4   13   15   24        32
5   12   13   30        30
9   10   17   36        36
3   25   26   36        54
7   15   20   42        42
10   13   13   60        36
8   15   17   60        40
...
Heronian triangles with area 210:
a    b    c area perimeter
---- ---- ---- ---- ---------
17   25   28  210        70
20   21   29  210        70
12   35   37  210        84
17   28   39  210        84
7   65   68  210       140
3  148  149  210       300


ALGOL W

Translation of: Lua
begin
% record to hold details of a Heronian triangle %
record Heronian ( integer a, b, c, area, perimeter );
% returns the details of the Heronian Triangle with sides a, b, c or nil if it isn't one %
reference(Heronian) procedure tryHt( integer value a, b, c ) ;
begin
real                s, areaSquared, area;
reference(Heronian) t;
s           := ( a + b + c ) / 2;
areaSquared := s * ( s - a ) * ( s - b ) * ( s - c );
t           := null;
if areaSquared > 0 then begin
% a, b, c does form a triangle %
area    := sqrt( areaSquared );
if entier( area ) = area then begin
% the area is integral so the triangle is Heronian %
t := Heronian( a, b, c, entier( area ), a + b + c )
end
end;
t
end tryHt ;

% returns the GCD of a and b %
integer procedure gcd( integer value a, b ) ; if b = 0 then a else gcd( b, a rem b );

% prints the details of the Heronian triangle t %
procedure htPrint( reference(Heronian) value t ) ; write( i_w := 4, s_w := 1, a(t), b(t), c(t), area(t), "     ", perimeter(t) );
% prints headings for the Heronian Triangle table %
procedure htTitle ; begin write( "   a    b    c area perimeter" ); write( "---- ---- ---- ---- ---------" ) end;

begin
% construct ht as a table of the Heronian Triangles with sides up to 200 %
reference(Heronian) array ht ( 1 :: 1000 );
reference(Heronian)       t;
integer                   htCount;

htCount := 0;
for c := 1 until 200 do begin
for b := 1 until c do begin
for a := 1 until b do begin
if gcd( gcd( a, b ), c ) = 1 then begin
t := tryHt( a, b, c );
if t not = null then begin
htCount       := htCount + 1;
ht( htCount ) := t
end
end
end
end
end;

% sort the table on ascending area, perimeter and max side length %
% note we constructed the triangles with c as the longest side %
begin
integer             lower, upper;
reference(Heronian) k, h;
logical             swapped;
lower := 1;
upper := htCount;
while begin
upper   := upper - 1;
swapped := false;
for i := lower until upper do begin
h := ht( i     );
k := ht( i + 1 );
if area(k) < area(h) or (   area(k) =  area(h)
and (  perimeter(k) <  perimeter(h)
or (   perimeter(k) = perimeter(h)
and c(k)         < c(h)
)
)
)
then begin
ht( i     ) := k;
ht( i + 1 ) := h;
swapped     := true;
end
end;
swapped
end
do  begin end;
end;

% display the triangles %
write( "There are ", htCount, " Heronian triangles with sides up to 200" );
htTitle;
for htPos := 1 until 10 do htPrint( ht( htPos ) );
write( " ..." );
write( "Heronian triangles with area 210:" );
htTitle;
for htPos := 1 until htCount do begin
reference(Heronian) t;
t := ht( htPos );
if area(t) = 210 then htPrint( t )
end
end
end.
Output:
There are            517   Heronian triangles with sides up to 200
a    b    c area perimeter
---- ---- ---- ---- ---------
3    4    5    6        12
5    5    6   12        16
5    5    8   12        18
4   13   15   24        32
5   12   13   30        30
9   10   17   36        36
3   25   26   36        54
7   15   20   42        42
10   13   13   60        36
8   15   17   60        40
...
Heronian triangles with area 210:
a    b    c area perimeter
---- ---- ---- ---- ---------
17   25   28  210        70
20   21   29  210        70
12   35   37  210        84
17   28   39  210        84
7   65   68  210       140
3  148  149  210       300


AppleScript

By composition of functional primitives, and using post-Yosemite AppleScript's ability to import Foundation classes (mainly for sorting records, here).

Translation of: JavaScript
use framework "Foundation"

-- HERONIAN TRIANGLES --------------------------------------------------------

-- heroniansOfSideUpTo :: Int -> [(Int, Int, Int)]
on heroniansOfSideUpTo(n)
script sideA
on |λ|(a)
script sideB
on |λ|(b)
script sideC
-- primitiveHeronian :: Int -> Int -> Int -> Bool
on primitiveHeronian(x, y, z)
(x ≤ y and y ≤ z) and (x + y > z) and ¬
gcd(gcd(x, y), z) = 1 and ¬
isIntegerValue(hArea(x, y, z))
end primitiveHeronian

on |λ|(c)
if primitiveHeronian(a, b, c) then
{{a, b, c}}
else
{}
end if
end |λ|
end script

concatMap(sideC, enumFromTo(b, n))
end |λ|
end script

concatMap(sideB, enumFromTo(a, n))
end |λ|
end script

concatMap(sideA, enumFromTo(1, n))
end heroniansOfSideUpTo

-- TEST ----------------------------------------------------------------------
on run
set n to 200

set lstHeron to ¬
sortByComparing({{"area", true}, {"perimeter", true}, {"maxSide", true}}, ¬
map(triangleDimensions, heroniansOfSideUpTo(n)))

set lstCols to {"sides", "perimeter", "area"}
set lstColWidths to {20, 15, 0}
set area to 210

script areaFilter
-- Record -> [Record]
on |λ|(recTriangle)
if area of recTriangle = area then
{recTriangle}
else
{}
end if
end |λ|
end script

intercalate("\n \n", {("Number of triangles found (with sides <= 200): " & ¬
length of lstHeron as string), ¬
¬
tabulation("First 10, ordered by area, perimeter, longest side", ¬
items 1 thru 10 of lstHeron, lstCols, lstColWidths), ¬
¬
tabulation("Area = 210", ¬
concatMap(areaFilter, lstHeron), lstCols, lstColWidths)})
end run

-- triangleDimensions :: (Int, Int, Int) ->
--       {sides: (Int, Int, Int),  area: Int, perimeter: Int, maxSize: Int}
on triangleDimensions(lstSides)
set {x, y, z} to lstSides
{sides:[x, y, z], area:hArea(x, y, z) as integer, perimeter:x + y + z, maxSide:z}
end triangleDimensions

-- hArea :: Int -> Int -> Int -> Num
on hArea(x, y, z)
set s to (x + y + z) / 2
set a to s * (s - x) * (s - y) * (s - z)

if a > 0 then
a ^ 0.5
else
0
end if
end hArea

-- gcd :: Int -> Int -> Int
on gcd(m, n)
if n = 0 then
m
else
gcd(n, m mod n)
end if
end gcd

-- TABULATION ----------------------------------------------------------------

-- tabulation :: [Record] -> [String] -> String -> [Integer] -> String
on tabulation(strLegend, lstRecords, lstKeys, lstWidths)
on |λ|(strTitle, iCol)
set str to toTitle(strTitle)
str & replicate((item iCol of lstWidths) - (length of str), space)
end |λ|
end script

script lineString
on |λ|(rec)
script fieldString
-- fieldString :: String -> Int -> String
on |λ|(strKey, i)
set v to keyValue(strKey, rec)

if class of v is list then
set strData to ("(" & intercalate(", ", v) & ")")
else
set strData to v as string
end if

strData & replicate(space, (item i of (lstWidths)) - (length of strData))
end |λ|
end script

tab & intercalate(tab, map(fieldString, lstKeys))
end |λ|
end script

strLegend & ":" & linefeed & linefeed & ¬
tab & intercalate(tab, ¬
map(heading, lstKeys)) & linefeed & ¬
intercalate(linefeed, map(lineString, lstRecords))
end tabulation

-- GENERIC FUNCTIONS ---------------------------------------------------------

-- concat :: [[a]] -> [a] | [String] -> String
on concat(xs)
if length of xs > 0 and class of (item 1 of xs) is string then
set acc to ""
else
set acc to {}
end if
repeat with i from 1 to length of xs
set acc to acc & item i of xs
end repeat
acc
end concat

-- concatMap :: (a -> [b]) -> [a] -> [b]
on concatMap(f, xs)
concat(map(f, xs))
end concatMap

-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m, n)
if m > n then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo

-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl

-- intercalate :: Text -> [Text] -> Text
on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate

-- isIntegerValue :: Num -> Bool
on isIntegerValue(n)
{real, integer} contains class of n and (n = (n as integer))
end isIntegerValue

-- keyValue :: String -> Record -> Maybe String
on keyValue(strKey, rec)
set ca to current application
set v to (ca's NSDictionary's dictionaryWithDictionary:rec)'s objectForKey:strKey
if v is not missing value then
item 1 of ((ca's NSArray's arrayWithObject:v) as list)
else
missing value
end if
end keyValue

-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map

-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn

-- replicate :: Int -> String -> String
on replicate(n, s)
set out to ""
if n < 1 then return out
set dbl to s

repeat while (n > 1)
if (n mod 2) > 0 then set out to out & dbl
set n to (n div 2)
set dbl to (dbl & dbl)
end repeat
return out & dbl
end replicate

-- List of {strKey, blnAscending} pairs -> list of records -> sorted list of records

-- sortByComparing :: [(String, Bool)] -> [Records] -> [Records]
on sortByComparing(keyDirections, xs)
set ca to current application

script recDict
on |λ|(x)
ca's NSDictionary's dictionaryWithDictionary:x
end |λ|
end script
set dcts to map(recDict, xs)

script asDescriptor
on |λ|(kd)
set {k, d} to kd
ca's NSSortDescriptor's sortDescriptorWithKey:k ascending:d selector:dcts
end |λ|
end script

((ca's NSArray's arrayWithArray:dcts)'s ¬
sortedArrayUsingDescriptors:map(asDescriptor, keyDirections)) as list
end sortByComparing

-- toTitle :: String -> String
on toTitle(str)
set ca to current application
((ca's NSString's stringWithString:(str))'s ¬
capitalizedStringWithLocale:(ca's NSLocale's currentLocale())) as text
end toTitle

Output:
Number of triangles found (with sides <= 200): 517

First 10, ordered by area, perimeter, longest side:

Sides                   Perimeter          Area
(3, 4, 5)               12                 6
(5, 5, 6)               16                 12
(5, 5, 8)               18                 12
(4, 13, 15)             32                 24
(5, 12, 13)             30                 30
(9, 10, 17)             36                 36
(3, 25, 26)             54                 36
(7, 15, 20)             42                 42
(10, 13, 13)            36                 60
(8, 15, 17)             40                 60

Area = 210:

Sides                   Perimeter          Area
(17, 25, 28)            70                 210
(20, 21, 29)            70                 210
(12, 35, 37)            84                 210
(17, 28, 39)            84                 210
(7, 65, 68)             140                210
(3, 148, 149)           300                210

Arturo

printTable: function [title, rows][
print title ++ ":"
print repeat "=" 60

print repeat "-" 60

loop rows 'row [
prints pad.center to :string row\0 10
prints pad.center to :string row\1 10
prints pad.center to :string row\2 10
prints pad.center to :string row\3 15
print pad.center to :string row\4 15
]
print ""
]

hero: function [a,b,c][
s: (a + b + c) // 2
return sqrt(s * (s-a) * (s-b) * (s-c))
]

heronian?: function [x]->
and? -> x > 0
-> x = ceil x

lst: []
mx: 200

loop 1..mx 'c ->
loop 1..c 'b ->
loop 1..b 'a [
area: hero a b c
if and? [heronian? area] [one? gcd @[a b c]]->
'lst ++ @[
@[a, b, c, a + b + c, to :integer area]
]
]

print ["Number of Heronian triangles:" size lst]
print ""

lst: arrange lst 'item ->
(item\4 * 10000) + (item\3 * 100) + max first.n:3 item

printTable "Ordered list of first ten Heronian triangles" first.n: 10 lst
printTable "Ordered list of Heronian triangles with area 210" select lst 'x -> x\4 = 210

Output:
Number of Heronian triangles: 517

Ordered list of first ten Heronian triangles:
============================================================
A         B         C        Perimeter        Area
------------------------------------------------------------
3         4         5           12              6
5         5         6           16             12
5         5         8           18             12
4         13        15          32             24
5         12        13          30             30
9         10        17          36             36
3         25        26          54             36
7         15        20          42             42
10        13        13          36             60
8         15        17          40             60

Ordered list of Heronian triangles with area 210:
============================================================
A         B         C        Perimeter        Area
------------------------------------------------------------
17        25        28          70             210
20        21        29          70             210
12        35        37          84             210
17        28        39          84             210
7         65        68          140            210
3        148       149          300            210

AutoHotkey

Primitive_Heronian_triangles(MaxSide){
obj :=[]
loop, % MaxSide {
a := A_Index
loop % MaxSide-a+1 {
b := A_Index+a-1
loop % MaxSide-b+1 {
c := A_Index+b-1, s := (a+b+c)/2, Area := Sqrt(s*(s-a)*(s-b)*(s-c))
if (Area = Floor(Area)) && (Area>0) && !obj[a/s, b/s, c/s]
obj[a/s, b/s, c/s]:=1 ,res .= (res?"n":"") StrReplace(Area, ".000000") "t" a+b+c "t" a ", " b ", " c
}	}	}
Sort, res, F Sort
return res
}

Sort(x, y){
x := StrSplit(x, "t"), y := StrSplit(y, "t")
return x.1 > y.1 ? 1 : x.1 < y.1 ? -1 : x.2 > y.2 ? 1 : x.2 < y.2 ? -1 : 0
}


Examples:

res := Primitive_Heronian_triangles(200)
loop, parse, res, n, r
{
if A_Index<=10
res2.= A_LoopField "n"
if StrSplit(A_LoopField, "t").1 = 210
res3.= A_LoopField "n"
Counter := A_Index
}

MsgBox % Counter " results found"
. "nnFirst 10 results:"
. "n" "AreatPerimetertSidesn" res2
. "nResults for Area = 210:"
. "n" "AreatPerimetertSidesn" res3
return


Outputs:

517 results found

First 10 results:
Area	Perimeter	Sides
6	12	3, 4, 5
12	16	5, 5, 6
12	18	5, 5, 8
24	32	4, 13, 15
30	30	5, 12, 13
36	36	9, 10, 17
36	54	3, 25, 26
42	42	7, 15, 20
60	36	10, 13, 13
60	40	8, 15, 17

Results for Area = 210:
Area	Perimeter	Sides
210	70	20, 21, 29
210	70	17, 25, 28
210	84	17, 28, 39
210	84	12, 35, 37
210	140	7, 65, 68
210	300	3, 148, 149

C

Takes max side, number of triangles to print and area limit as inputs. Area should be -1 if it is not a restriction. Triangles are stored in a linked list which is built sorted and hence no post processing is required. Usage is printed out on incorrect invocation.

IMPORTANT: This is a C99 compatible implementation. May result in errors on earlier compilers.

#include<stdlib.h>
#include<stdio.h>
#include<math.h>

typedef struct{
int a,b,c;
int perimeter;
double area;
}triangle;

typedef struct elem{
triangle t;
struct elem* next;
}cell;

typedef cell* list;

list iter,temp;
int flag = 0;

if(*a==NULL){
*a = (list)malloc(sizeof(cell));
(*a)->t = t;
(*a)->next = NULL;
}

else{
temp = (list)malloc(sizeof(cell));

iter = *a;
while(iter->next!=NULL){
if(((iter->t.area<t.area)||(iter->t.area==t.area && iter->t.perimeter<t.perimeter)||(iter->t.area==t.area && iter->t.perimeter==t.perimeter && iter->t.a<=t.a))
&&
(iter->next==NULL||(t.area<iter->next->t.area || t.perimeter<iter->next->t.perimeter || t.a<iter->next->t.a))){
temp->t = t;
temp->next = iter->next;
iter->next = temp;
flag = 1;
break;
}

iter = iter->next;
}

if(flag!=1){
temp->t = t;
temp->next = NULL;
iter->next = temp;
}
}
}

int gcd(int a,int b){
if(b!=0)
return gcd(b,a%b);
return a;
}

void calculateArea(triangle *t){
(*t).perimeter = (*t).a + (*t).b + (*t).c;
(*t).area = sqrt(0.5*(*t).perimeter*(0.5*(*t).perimeter - (*t).a)*(0.5*(*t).perimeter - (*t).b)*(0.5*(*t).perimeter - (*t).c));
}

list generateTriangleList(int maxSide,int *count){
int a,b,c;
triangle t;
list herons = NULL;

*count = 0;

for(a=1;a<=maxSide;a++){
for(b=1;b<=a;b++){
for(c=1;c<=b;c++){
if(c+b > a && gcd(gcd(a,b),c)==1){
t = (triangle){a,b,c};
calculateArea(&t);
if(t.area/(int)t.area == 1){
(*count)++;
}
}
}
}
}

return herons;
}

void printList(list a,int limit,int area){
list iter = a;
int count = 1;

printf("\nDimensions\tPerimeter\tArea");

while(iter!=NULL && count!=limit+1){
if(area==-1 ||(area==iter->t.area)){
printf("\n%d x %d x %d\t%d\t\t%d",iter->t.a,iter->t.b,iter->t.c,iter->t.perimeter,(int)iter->t.area);
count++;
}
iter = iter->next;
}
}

int main(int argC,char* argV[])
{
int count;
list herons = NULL;

if(argC!=4)
printf("Usage : %s <Max side, max triangles to print and area, -1 for area to ignore>",argV[0]);
else{
herons = generateTriangleList(atoi(argV[1]),&count);
printf("Triangles found : %d",count);
(atoi(argV[3])==-1)?printf("\nPrinting first %s triangles.",argV[2]):printf("\nPrinting triangles with area %s square units.",argV[3]);
printList(herons,atoi(argV[2]),atoi(argV[3]));
free(herons);
}
return 0;
}


Invocation and output :

C:\rosettaCode>heronian.exe 200 10 -1
Triangles found : 517
Printing first 10 triangles.
Dimensions      Perimeter       Area
5 x 4 x 3       12              6
6 x 5 x 5       16              12
8 x 5 x 5       18              12
15 x 13 x 4     32              24
13 x 12 x 5     30              30
17 x 10 x 9     36              36
26 x 25 x 3     54              36
20 x 15 x 7     42              42
13 x 13 x 10    36              60
17 x 15 x 8     40              60
C:\rosettaCode>heronian.exe 200 10 210
Triangles found : 517
Printing triangles with area 210 square units.
Dimensions      Perimeter       Area
28 x 25 x 17    70              210
29 x 21 x 20    70              210
37 x 35 x 12    84              210
39 x 28 x 17    84              210
68 x 65 x 7     140             210
149 x 148 x 3   300             210


C#

using System;
using System.Collections.Generic;

namespace heron
{
class Program{
static void Main(string[] args){
List<int[]> list = new List<int[]>();
for (int c = 1; c <= 200; c++)
for (int b = 1; b <= c; b++)
for (int a = 1; a <= b; a++)
if (gcd(a, gcd(b, c)) == 1 && isHeron(heronArea(a, b, c)))
list.Add(new int[] { a, b, c, a + b + c, (int)heronArea(a, b, c)});
sort(list);
Console.WriteLine("Number of primitive Heronian triangles with sides up to 200: " + list.Count + "\n\nFirst ten when ordered by increasing area, then perimeter,then maximum sides:\nSides\t\t\tPerimeter\tArea");
for(int i = 0; i < 10; i++)
Console.WriteLine(list[i][0] + "\t" + list[i][1] + "\t" + list[i][2] + "\t" + list[i][3] + "\t\t" + list[i][4]);
Console.WriteLine("\nPerimeter = 210\nSides\t\t\tPerimeter\tArea");
foreach (int[] i in list)
if (i[4] == 210)
Console.WriteLine(i[0] + "\t" + i[1] + "\t" + i[2] + "\t" + i[3] + "\t\t" + i[4]);
}
static bool isHeron(double heronArea){
return heronArea % 1 == 0 && heronArea != 0;
}
static double heronArea(int a, int b, int c){
double s = (a + b + c) / 2d;
return Math.Sqrt(s * (s - a) * (s - b) * (s - c));
}
static int gcd(int a, int b){
int remainder = 1, dividend, divisor;
dividend = a > b ? a : b;
divisor = a > b ? b : a;
while (remainder != 0){
remainder = dividend % divisor;
if (remainder != 0){
dividend = divisor;
divisor = remainder;
}
}
return divisor;
}
static void sort(List<int[]> list){
int[] temp = new int[5];
bool changed = true;
while(changed){
changed = false;
for (int i = 1; i < list.Count; i++)
if (list[i][4] < list[i - 1][4] || list[i][4] == list[i - 1][4] && list[i][3] < list[i - 1][3]){
temp = list[i];
list[i] = list[i - 1];
list[i - 1] = temp;
changed = true;
}
}
}
}
}

Output:
Number of primitive Heronian triangles with sides up to 200: 517

First ten when ordered by increasing area, then perimeter,then maximum sides:
Sides                   Perimeter       Area
3       4       5       12              6
5       5       6       16              12
5       5       8       18              12
4       13      15      32              24
5       12      13      30              30
9       10      17      36              36
3       25      26      54              36
7       15      20      42              42
10      13      13      36              60
8       15      17      40              60

Perimeter = 210
Sides                   Perimeter       Area
17      25      28      70              210
20      21      29      70              210
12      35      37      84              210
17      28      39      84              210
7       65      68      140             210
3       148     149     300             210

C++

Works with: C++17
#include <tuple>
#include <vector>
#include <numeric>
#include <iostream>
#include <algorithm>

#include <cmath>

struct Triangle {
int a{};
int b{};
int c{};

[[nodiscard]] constexpr auto perimeter() const noexcept { return a + b + c; }

[[nodiscard]] constexpr auto area() const noexcept {
const auto p_2 = static_cast<double>(perimeter()) / 2;
const auto area_sq = p_2 * (p_2 - a) * (p_2 - b) * (p_2 - c);
return std::sqrt(area_sq);
}
};

auto generate_triangles(int side_limit = 200) {
std::vector<Triangle> result;
for(int a = 1; a <= side_limit; ++a)
for(int b = 1; b <= a; ++b)
for(int c = a + 1 - b; c <= b; ++c) // skip too-small values of c, which will violate triangle inequality
{
Triangle t{ a, b, c };
const auto t_area = t.area();
if (t_area == 0) continue;
if (std::floor(t_area) == std::ceil(t_area) && std::gcd(a, std::gcd(b, c)) == 1)
result.push_back(t);
}
return result;
}

bool compare(const Triangle& lhs, const Triangle& rhs) noexcept {
return std::make_tuple(lhs.area(), lhs.perimeter(), std::max(lhs.a, std::max(lhs.b, lhs.c))) <
std::make_tuple(rhs.area(), rhs.perimeter(), std::max(rhs.a, std::max(rhs.b, rhs.c)));
}

struct area_compare {
[[nodiscard]] constexpr bool operator()(const Triangle& t, int i) const noexcept { return t.area() < i; }
[[nodiscard]] constexpr bool operator()(int i, const Triangle& t) const noexcept { return i < t.area(); }
};

int main() {
auto tri = generate_triangles();
std::cout << "There are " << tri.size() << " primitive Heronian triangles with sides up to 200\n\n";

std::cout << "First ten when ordered by increasing area, then perimeter, then maximum sides:\n";
std::sort(tri.begin(), tri.end(), compare);
std::cout << "area\tperimeter\tsides\n";
for(int i = 0; i < 10; ++i)
std::cout << tri[i].area() << '\t' << tri[i].perimeter() << "\t\t" <<
tri[i].a << 'x' << tri[i].b << 'x' << tri[i].c << '\n';

std::cout << "\nAll with area 210 subject to the previous ordering:\n";
auto range = std::equal_range(tri.begin(), tri.end(), 210, area_compare());
std::cout << "area\tperimeter\tsides\n";
for(auto it = range.first; it != range.second; ++it)
std::cout << (*it).area() << '\t' << (*it).perimeter() << "\t\t" <<
it->a << 'x' << it->b << 'x' << it->c << '\n';
}

Output:
There are 517 primitive Heronian triangles with sides up to 200

First ten when ordered by increasing area, then perimeter, then maximum sides:
area    perimeter       sides
6       12              5x4x3
12      16              6x5x5
12      18              8x5x5
24      32              15x13x4
30      30              13x12x5
36      36              17x10x9
36      54              26x25x3
42      42              20x15x7
60      36              13x13x10
60      40              17x15x8

All with area 210 subject to the previous ordering:
area    perimeter       sides
210     70              28x25x17
210     70              29x21x20
210     84              37x35x12
210     84              39x28x17
210     140             68x65x7
210     300             149x148x3

CoffeeScript

Translation of: JavaScript
heronArea = (a, b, c) ->
s = (a + b + c) / 2
Math.sqrt s * (s - a) * (s - b) * (s - c)

isHeron = (h) -> h % 1 == 0 and h > 0

gcd = (a, b) ->
leftover = 1
dividend = if a > b then a else b
divisor = if a > b then b else a
until leftover == 0
leftover = dividend % divisor
if leftover > 0
dividend = divisor
divisor = leftover
divisor

list = []
for c in [1..200]
for b in [1..c]
for a in [1..b]
area = heronArea(a, b, c)
if gcd(gcd(a, b), c) == 1 and isHeron(area)
list.push new Array(a, b, c, a + b + c, area)

sort = (list) ->
swapped = true
while swapped
swapped = false
for i in [1..list.length-1]
if list[i][4] < list[i - 1][4] or list[i][4] == list[i - 1][4] and list[i][3] < list[i - 1][3]
temp = list[i]
list[i] = list[i - 1]
list[i - 1] = temp
swapped = true
sort list

# some results:
console.log 'primitive Heronian triangles with sides up to 200: ' + list.length
console.log 'First ten when ordered by increasing area, then perimeter:'
for i in list[0..10-1]
console.log  i[0..2].join(' x ') + ', p = ' + i[3] + ', a = ' + i[4]

console.log '\nHeronian triangles with area = 210:'
for i in list
if i[4] == 210
console.log i[0..2].join(' x ') + ', p = ' + i[3]

Output:
primitive Heronian triangles with sides up to 200: 517
First ten when ordered by increasing area, then perimeter:
3 x 4 x 5, p = 12, a = 6
5 x 5 x 6, p = 16, a = 12
5 x 5 x 8, p = 18, a = 12
4 x 13 x 15, p = 32, a = 24
5 x 12 x 13, p = 30, a = 30
9 x 10 x 17, p = 36, a = 36
3 x 25 x 26, p = 54, a = 36
7 x 15 x 20, p = 42, a = 42
10 x 13 x 13, p = 36, a = 60
8 x 15 x 17, p = 40, a = 60

Heronian triangles with area = 210:
17 x 25 x 28, p = 70
20 x 21 x 29, p = 70
12 x 35 x 37, p = 84
17 x 28 x 39, p = 84
7 x 65 x 68, p = 140
3 x 148 x 149, p = 300

D

Translation of: Python
import std.stdio, std.math, std.range, std.algorithm, std.numeric, std.traits, std.typecons;

double hero(in uint a, in uint b, in uint c) pure nothrow @safe @nogc {
immutable s = (a + b + c) / 2.0;
immutable a2 = s * (s - a) * (s - b) * (s - c);
return (a2 > 0) ? a2.sqrt : 0.0;
}

bool isHeronian(in uint a, in uint b, in uint c) pure nothrow @safe @nogc {
immutable h = hero(a, b, c);
return h > 0 && h.floor == h.ceil;
}

T gcd3(T)(in T x, in T y, in T z) pure nothrow @safe @nogc {
return gcd(gcd(x, y), z);
}

void main() /*@safe*/ {
enum uint maxSide = 200;

// Sort by increasing area, perimeter, then sides.
//auto h = cartesianProduct!3(iota(1, maxSide + 1))
auto r = iota(1, maxSide + 1);
const h = cartesianProduct(r, r, r)
//.filter!({a, b, c} => ...
.filter!(t => t[0] <= t[1] && t[1] <= t[2] &&
t[0] + t[1] > t[2] &&
t[].gcd3 == 1 && t[].isHeronian)
.array
.schwartzSort!(t => tuple(t[].hero, t[].only.sum, t.reverse))
.release;

static void showTriangles(R)(R ts) @safe {
"Area Perimeter Sides".writeln;
foreach (immutable t; ts)
writefln("%3s %8d %3dx%dx%d", t[].hero, t[].only.sum, t[]);
}

writefln("Primitive Heronian triangles with sides up to %d: %d", maxSide, h.length);
"\nFirst ten when ordered by increasing area, then perimeter,then maximum sides:".writeln;
showTriangles(h.take(10));

"\nAll with area 210 subject to the previous ordering:".writeln;
showTriangles(h.filter!(t => t[].hero == 210));
}

Output:
Primitive Heronian triangles with sides up to 200: 517

First ten when ordered by increasing area, then perimeter,then maximum sides:
Area Perimeter Sides
6       12   3x4x5
12       16   5x5x6
12       18   5x5x8
24       32   4x13x15
30       30   5x12x13
36       36   9x10x17
36       54   3x25x26
42       42   7x15x20
60       36  10x13x13
60       40   8x15x17

All with area 210 subject to the previous ordering:
Area Perimeter Sides
210       70  17x25x28
210       70  20x21x29
210       84  12x35x37
210       84  17x28x39
210      140   7x65x68
210      300   3x148x149

See Pascal.

EasyLang

func gcd x y .
if y = 0
return x
.
return gcd y (x mod y)
.
global ta[] tb[] tc[] .
proc mktbl . .
for c = 1 to 200
for b = 1 to c
for a = 1 to b
s = (a + b + c) / 2
ar = sqrt (s * (s - a) * (s - b) * (s - c))
if ar > 0 and ar = floor ar
if gcd gcd b c a = 1
ta[] &= a
tb[] &= b
tc[] &= c
.
.
.
.
.
.
mktbl
#
proc get i . a b c per ar .
a = ta[i]
b = tb[i]
c = tc[i]
per = a + b + c
s = per / 2
ar = sqrt (s * (s - a) * (s - b) * (s - c))
.
func wgt i .
get i a b c per ar
return ar * 1000000 + per * 1000 + a
.
proc sort . .
for i = 1 to len ta[] - 1
for j = i + 1 to len ta[]
if wgt j < wgt i
swap ta[j] ta[i]
swap tb[j] tb[i]
swap tc[j] tc[i]
.
.
.
.
sort
#
print "Number of triangles: " & len ta[]
print ""
numfmt 2 2
for i = 1 to 10
get i a b c per ar
print "(" & a & " " & b & " " & c & ") Perim: " & per & " Area: " & ar
.
print ""
for i = 1 to len ta[]
get i a b c per ar
if ar = 210
print "(" & a & " " & b & " " & c & ") Perim: " & per & " Area: " & ar
.
.

Output:
Number of triangles: 517

( 3  4  5) Perim: 12 Area:  6
( 5  5  6) Perim: 16 Area: 12
( 5  5  8) Perim: 18 Area: 12
( 4 13 15) Perim: 32 Area: 24
( 5 12 13) Perim: 30 Area: 30
( 9 10 17) Perim: 36 Area: 36
( 3 25 26) Perim: 54 Area: 36
( 7 15 20) Perim: 42 Area: 42
(10 13 13) Perim: 36 Area: 60
( 8 15 17) Perim: 40 Area: 60

(17 25 28) Perim: 70 Area: 210
(20 21 29) Perim: 70 Area: 210
(12 35 37) Perim: 84 Area: 210
(17 28 39) Perim: 84 Area: 210
( 7 65 68) Perim: 140 Area: 210
( 3 148 149) Perim: 300 Area: 210


EchoLisp

;; returns quintuple (A s a b c)
;; or #f if not hero
(define (hero a b c (s 0) (A 0))
(when
(= 1 (gcd a b c))
(set! s (// (+ a b c) 2))
(set! A (* s (- s a)(- s b)(- s c)))
(when (square? A)
(list (sqrt A) (* s 2)  c b a))))

;; all heroes a,b,c < sidemax
;; sorted by A|s|c & a <=b <= c
(define (heroes (sidemax 201))
(list-sort/fields 3
(for*/list ((a (in-range 1 sidemax)) (b (in-range a sidemax)) (c (in-range b sidemax)))
#:continue (<= (+ a b) c) ;; triangle inequality must hold !! cut search
#:continue (not (hero a b c))
(hero a b c))))

(define (print-hero h)
(printf "A: %6d s: %6d    sides: %dx%dx%d"
(list-ref  h  0) (list-ref  h  1)
(list-ref h 2)(list-ref h 3) (list-ref h 4)))
(define (print-laurels H)
(writeln '🌿🌿 (length H) 'heroes '🌿🌿))

Output:
(define H (heroes))

(print-laurels H)
🌿🌿     517     heroes     🌿🌿

(for-each print-hero (take H 10))

A:      6 s:     12 sides: 5x4x3
A:     12 s:     16 sides: 6x5x5
A:     12 s:     18 sides: 8x5x5
A:     24 s:     32 sides: 15x13x4
A:     30 s:     30 sides: 13x12x5
A:     36 s:     36 sides: 17x10x9
A:     36 s:     54 sides: 26x25x3
A:     42 s:     42 sides: 20x15x7
A:     60 s:     36 sides: 13x13x10
A:     60 s:     40 sides: 17x15x8

(for-each print-hero (filter (lambda(h) (= 210 (first h))) H))

A:    210 s:     70 sides: 28x25x17
A:    210 s:     70 sides: 29x21x20
A:    210 s:     84 sides: 37x35x12
A:    210 s:     84 sides: 39x28x17
A:    210 s:    140 sides: 68x65x7
A:    210 s:    300 sides: 149x148x3

Elixir

defmodule Heronian do
def triangle?(a,b,c) when a+b <= c, do: false
def triangle?(a,b,c) do
area = area(a,b,c)
area == round(area) and primitive?(a,b,c)
end

def area(a,b,c) do
s = (a + b + c) / 2
:math.sqrt(s * (s-a) * (s-b) * (s-c))
end

defp primitive?(a,b,c), do: gcd(gcd(a,b),c) == 1

defp gcd(a,0), do: a
defp gcd(a,b), do: gcd(b, rem(a,b))
end

max = 200
triangles = for a <- 1..max, b <- a..max, c <- b..max, Heronian.triangle?(a,b,c), do: {a,b,c}
IO.puts length(triangles)

IO.puts "\nSides\t\t\tPerim\tArea"
Enum.map(triangles, fn {a,b,c} -> {Heronian.area(a,b,c),a,b,c} end)
|> Enum.sort
|> Enum.take(10)
|> Enum.each(fn {area, a, b, c} ->
IO.puts "#{a}\t#{b}\t#{c}\t#{a+b+c}\t#{round(area)}"
end)
IO.puts ""
area_size = 210
Enum.filter(triangles, fn {a,b,c} -> Heronian.area(a,b,c) == area_size end)
|> Enum.sort_by(fn {a,b,c} -> a+b+c end)
|> Enum.each(fn {a, b, c} ->
IO.puts "#{a}\t#{b}\t#{c}\t#{a+b+c}\t#{area_size}"
end)

Output:
517

Sides                   Perim   Area
3       4       5       12      6
5       5       6       16      12
5       5       8       18      12
4       13      15      32      24
5       12      13      30      30
3       25      26      54      36
9       10      17      36      36
7       15      20      42      42
6       25      29      60      60
8       15      17      40      60

17      25      28      70      210
20      21      29      70      210
12      35      37      84      210
17      28      39      84      210
7       65      68      140     210
3       148     149     300     210

ERRE

PROGRAM HERON

DIM LISTA%[600,4]

PROCEDURE GCD(J%,K%->MCD%)
WHILE J%<>K% DO
IF J%>K% THEN
J%=J%-K%
ELSE
K%=K%-J%
END IF
END WHILE
MCD%=J%
END PROCEDURE

BEGIN
PRINT(CHR$(12);) !CLS FOR C%=1 TO 200 DO FOR B%=1 TO C% DO FOR A%=1 TO B% DO S#=(A%+B%+C%)/2# AREA#=S#*(S#-A%)*(S#-B%)*(S#-C%) IF AREA#>0 THEN AREA#=SQR(AREA#) IF AREA#=INT(AREA#) THEN GCD(B%,C%->RES%) GCD(A%,RES%->RES%) IF RES%=1 THEN COUNT%=COUNT%+1 LISTA%[COUNT%,0]=A% LISTA%[COUNT%,1]=B% LISTA%[COUNT%,2]=C% LISTA%[COUNT%,3]=2*S# LISTA%[COUNT%,4]=AREA# END IF END IF END IF END FOR END FOR END FOR PRINT("Number of triangles:";COUNT%) ! sorting array FLIPS%=TRUE WHILE FLIPS% DO FLIPS%=FALSE FOR I%=1 TO COUNT%-1 DO IF LISTA%[I%,4]>LISTA%[I%+1,4] THEN FOR K%=0 TO 4 DO SWAP(LISTA%[I%,K%],LISTA%[I%+1,K%]) END FOR FLIPS%=TRUE END IF END FOR END WHILE ! first ten FOR I%=1 TO 10 DO PRINT(#1,LISTA%[I%,0],LISTA%[I%,1],LISTA%[I%,2],LISTA%[I%,3],LISTA%[I%,4]) END FOR PRINT ! triangle with area=210 FOR I%=1 TO COUNT% DO IF LISTA%[I%,4]=210 THEN PRINT(LISTA%[I%,0],LISTA%[I%,1],LISTA%[I%,2],LISTA%[I%,3],LISTA%[I%,4]) END IF END FOR END PROGRAM Number of triangles: 517 3 4 5 12 6 5 5 6 16 12 5 5 8 18 12 4 13 15 32 24 5 12 13 30 30 9 10 17 36 36 3 25 26 54 36 7 15 20 42 42 10 13 13 36 60 8 15 17 40 60 17 25 28 70 210 20 21 29 70 210 12 35 37 84 210 17 28 39 84 210 7 65 68 140 210 3 148 149 300 210 Factor USING: accessors assocs backtrack combinators.extras combinators.short-circuit formatting io kernel locals math math.functions math.order math.parser math.ranges mirrors qw sequences sorting.slots ; IN: rosetta-code.heronian-triangles TUPLE: triangle a b c area perimeter ; :: area ( a b c -- x ) a b + c + 2 / :> s s s a - * s b - * s c - * sqrt ; : <triangle> ( triplet-seq -- triangle ) [ first3 ] [ first3 area >integer ] [ sum ] tri triangle boa ; : heronian? ( a b c -- ? ) area dup [ complex? ] [ 0 number= ] bi or [ drop f ] [ dup >integer number= ] if ; : 3gcd ( a b c -- n ) [ gcd nip ] twice ; : primitive-heronian? ( a b c -- ? ) { [ 3gcd 1 = ] [ heronian? ] } 3&& ; :: find-triangles ( -- seq ) [ 200 [1,b] amb-lazy :> c ! Use backtrack vocab to test c [1,b] amb-lazy :> b ! permutations of sides such b [1,b] amb-lazy :> a ! that c >= b >= a. a b c primitive-heronian? must-be-true { a b c } <triangle> ] bag-of ; ! collect every triangle : sort-triangles ( seq -- seq' ) { { area>> <=> } { perimeter>> <=> } } sort-by ; CONSTANT: format "%4s%5s%5s%5s%10s\n" : print-header ( -- ) qw{ a b c area perimeter } format vprintf "---- ---- ---- ---- ---------" print ; : print-triangle ( triangle -- ) <mirror> >alist values [ number>string ] map format vprintf ; : print-triangles ( seq -- ) [ print-triangle ] each ; inline : first10 ( sorted-triangles -- ) dup length "%d triangles found. First 10: \n" printf print-header 10 head print-triangles ; : area210= ( sorted-triangles -- ) "Triangles with area 210: " print print-header [ area>> 210 = ] filter print-triangles ; : main ( -- ) "Finding heronian triangles with sides <= 200..." print nl find-triangles sort-triangles [ first10 nl ] [ area210= ] bi ; MAIN: main  Output: Finding heronian triangles with sides <= 200... 517 triangles found. First 10: a b c area perimeter ---- ---- ---- ---- --------- 3 4 5 6 12 5 5 6 12 16 5 5 8 12 18 4 13 15 24 32 5 12 13 30 30 9 10 17 36 36 3 25 26 36 54 7 15 20 42 42 10 13 13 60 36 8 15 17 60 40 Triangles with area 210: a b c area perimeter ---- ---- ---- ---- --------- 17 25 28 210 70 20 21 29 210 70 12 35 37 210 84 17 28 39 210 84 7 65 68 210 140 3 148 149 210 300  Fortran Earlier Fortran doesn't offer special functions such as SUM, PRODUCT and MAXVAL of arrays, nor the ability to create compound data aggregates such as STASH to store a triangle's details. Simple code would have to be used in the absence of such conveniences, and multiple ordinary arrays rather than an array of a compound data entity. Rather than attempt to create the candidate triangles in the desired order, the simple approach is to sort a list of triangles, and using an XNDX array evades tossing compound items about. Rather than create a procedure to do the sorting, a comb sort is not too much trouble to place in-line once. Further, since the ordering is based on a compound key, having only one comparison to code is a boon. The three-way-if statement is central to the expedient evaluation of a compound sort key, but this facility is deprecated by the modernists, with no alternative offered that avoids re-comparison of parts.  MODULE GREEK MATHEMATICIANS !Two millenia back and more. CONTAINS INTEGER FUNCTION GCD(I,J) !Greatest common divisor. INTEGER I,J !Of these two integers. INTEGER N,M,R !Workers. N = MAX(I,J) !Since I don't want to damage I or J, M = MIN(I,J) !These copies might as well be the right way around. 1 R = MOD(N,M) !Divide N by M to get the remainder R. c write (6,*) "M,N,R",M,N,R IF (R.GT.0) THEN !Remainder zero? N = M !No. Descend a level. M = R !M-multiplicity has been removed from N. IF (R .GT. 1) GO TO 1 !No point dividing by one. END IF !If R = 0, M divides N. GCD = M !There we are. END FUNCTION GCD !Euclid lives on! FUNCTION GCD3(I,J,K) !Double do. INTEGER I,J,K !Three numbers. INTEGER R !One remainder. R = GCD(I,J) !Greatest common divisor. IF (R .GT. 1) R = GCD(R,K) !The first two might be co-prime. GCD3 = R !The result. END FUNCTION GCD3 REAL*8 FUNCTION HERO(SIDE) !Hero's calculation for the area of a triangle. Calculations could proceed with non-integer sides. INTEGER SIDE(3) !The lengths of each of the sides. REAL*8 S !A scratchpad. S = SUM(SIDE) !Definitely integer arithmetic. S = S/2 !Full precision without muttering /2D0. S = S*PRODUCT(S - SIDE) !Negative for non-joining triangles. HERO = SIGN(SQRT(ABS(S)),S) !Protect the SQRT against such. END FUNCTION HERO !As when one side is longer than the other two combined. END MODULE GREEK MATHEMATICIANS !Only a selection here. PROGRAM TEST !Find triangles with integral sides and areas. USE GREEK MATHEMATICIANS !For guidance. INTEGER LIMIT,LOTS !And then descend to Furrytran. PARAMETER (LIMIT = 200, LOTS = 666) !This should do. INTEGER I,J,K,SIDE(3) !The lengths of the sides of the triangles. EQUIVALENCE (SIDE(1),I),(SIDE(2),J),(SIDE(3),K) !I want two access styles. REAL*8 A !The area of the triangle. TYPE ABLOB !Define a stash for the desired results. INTEGER SIDE(3) !The three sides, INTEGER PERIMETER !Their summation, somewhat redundant. INTEGER AREA !This is rather more difficult to calculate. END TYPE ABLOB !That will do. TYPE(ABLOB) STASH(LOTS) !I'll have some. INTEGER N,XNDX(LOTS) !A counter and an index.. INTEGER H,T !Stuff for the in-line combsort. LOGICAL CURSE !Rather than mess with subroutines and parameters. INTEGER TASTE,CHOICE !Output selection stuff. PARAMETER (TASTE = 10, CHOICE = 210) !As specified. Collect some triangles. N = 0 !So, here we go. DO K = 1,LIMIT !Just slog away, DO J = 1,K !With brute force and ignorance. DO I = 1,J !This way, a 3,4,5 triangle is in that order. IF (GCD3(I,J,K).GT.1) CYCLE !A mere multiple. Seen it before. A = HERO(SIDE) !Assess the area. IF (A.LE.0) CYCLE !Not a valid triangle! IF (A .NE. INT(A)) CYCLE !Not an integral area. Precision is adequate... N = N + 1 !Another candidate survives. IF (N.GT.LOTS) STOP "Too many!" !Perhaps not for long! XNDX(N) = N !So, keep a finger. STASH(N).SIDE = SIDE !Stash its details. STASH(N).PERIMETER = SUM(SIDE) !Calculate once, here. STASH(N).AREA = A !And save this as an integer. c WRITE (6,10) N,STASH(N) 10 FORMAT (I4,":",3I4,I7,I8) !A reasonable layout. END DO END DO END DO WRITE (6,11) N,LIMIT !The first result. 11 FORMAT (I0," triangles of integral area. Sides up to ",I0) Comb sort involves coding only one test, and the comparison is to be compound... H = N - 1 !Last - First, and not +1. 20 H = MAX(1,H*10/13) !The special feature. IF (H.EQ.9 .OR. H.EQ.10) H = 11 !A twiddle. CURSE = .FALSE. !So far, so good. DO 24 I = N - H,1,-1 !If H = 1, this is a BubbleSort. IF (STASH(XNDX(I)).AREA - STASH(XNDX(I + H)).AREA) 24,21,23 !One compare. But, a compound key. 21 IF (STASH(XNDX(I)).PERIMETER-STASH(XNDX(I+H)).PERIMETER)24,22,23 !Equal area, so, perimeter? 22 IF (MAXVAL(STASH(XNDX(I)).SIDE) !Equal perimeter, so, longest side? 1 - MAXVAL(STASH(XNDX(I+H)).SIDE)) 24,24,23 !At last, equality here can be passed over. 23 T=XNDX(I); XNDX(I)=XNDX(I+H); XNDX(I+H)=T !One swap. CURSE = .TRUE. !One curse. 24 CONTINUE !One loop. IF (CURSE .OR. H.GT.1) GO TO 20 !Work remains? Cast forth the results, as per the specification. WRITE (6,30) TASTE 30 FORMAT ("First ",I0,", ordered by area, perimeter, longest side.", 1 /,"Index ---Sides--- Perimeter Area") DO I = 1,TASTE WRITE (6,10) XNDX(I),STASH(XNDX(I)) END DO WRITE (6,31) CHOICE 31 FORMAT ("Those triangles with area",I7) DO I = 1,N !I could go looking through the ordered list for CHOICE entries, IF (STASH(XNDX(I)).AREA.NE.CHOICE) CYCLE!But I can't be bothered. WRITE (6,10) XNDX(I),STASH(XNDX(I)) !Here is one such. END DO !Just thump through the lot. END  Output: 517 triangles of integral area. Sides up to 200 First 10, ordered by area, perimeter, longest side. Index ---Sides--- Perimeter Area 1: 3 4 5 12 6 2: 5 5 6 16 12 3: 5 5 8 18 12 6: 4 13 15 32 24 4: 5 12 13 30 30 8: 9 10 17 36 36 19: 3 25 26 54 36 12: 7 15 20 42 42 5: 10 13 13 36 60 9: 8 15 17 40 60 Those triangles with area 210 21: 17 25 28 70 210 22: 20 21 29 70 210 33: 12 35 37 84 210 36: 17 28 39 84 210 91: 7 65 68 140 210 329: 3 148 149 300 210 FreeBASIC ' version 02-05-2016 ' compile with: fbc -s console #Macro header Print Print " a b c s area" Print "-----------------------------------" #EndMacro Type triangle Dim As UInteger a Dim As UInteger b Dim As UInteger c Dim As UInteger s Dim As UInteger area End Type Function gcd(x As UInteger, y As UInteger) As UInteger Dim As UInteger t While y t = y y = x Mod y x = t Wend Return x End Function Function Heronian_triangles(a_max As UInteger, b_max As UInteger, _ c_max As UInteger, result() As triangle) As UInteger Dim As UInteger a, b, c Dim As UInteger s, sqroot, total, temp For a = 1 To a_max For b = a To b_max ' make sure that a + b + c is even For c = b + (a And 1) To c_max Step 2 ' to form a triangle a + b must be greater then c If (a + b) <= c Then Exit For ' check if a, b and c have a common divisor If (gcd(c, b) <> 1 And gcd(c, a) <> 1) Then Continue For End If s = (a + b + c) \ 2 temp = s * (s - a) * (s - b) * (s - c) sqroot = Sqr(temp) If (sqroot * sqroot) = temp Then total += 1 With result(total) .a = a .b = b .c = c .s = s .area = sqroot End With End If Next Next Next Return total End Function Sub sort_tri(result() As triangle, total As UInteger) ' shell sort ' sort order: area, s, c Dim As UInteger x, y, inc, done inc = total Do inc = IIf(inc > 1, inc \ 2, 1) Do done = 0 For x = 1 To total - inc y = x + inc If result(x).area > result(y).area Then Swap result(x), result(y) done = 1 Else If result(x).area = result(y).area Then If result(x).s > result(y).s Then Swap result(x), result(y) done = 1 Else If result(x).s = result(y).s Then If result(x).c > result(y).c Then Swap result(x), result(y) done = 1 End If End If End If End If End If Next Loop Until done = 0 Loop Until inc = 1 End Sub ' ------=< MAIN >=------ ReDim result(1 To 1000) As triangle Dim As UInteger x, y, total total = Heronian_triangles(200, 200, 200, result() ) ' trim the array by removing empty entries ReDim Preserve result(1 To total ) As triangle sort_tri(result(), total) Print "There are ";total;" Heronian triangles with sides <= 200" Print Print "First ten sorted entries" header ' print header For x = 1 To IIf(total > 9, 10, total) With result(x) Print Using " #####"; .a; .b; .c; .s; .area End With Next Print Print Print "Entries with a area = 210" header ' print header For x = 1 To UBound(result) With result(x) If .area = 210 Then Print Using " #####"; .a; .b; .c; .s; .area End If End With Next ' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End  Output: There are 517 Heronian triangles with sides <= 200 First ten sorted entries a b c s area ----------------------------------- 3 4 5 6 6 5 5 6 8 12 5 5 8 9 12 4 13 15 16 24 5 12 13 15 30 9 10 17 18 36 3 25 26 27 36 7 15 20 21 42 10 13 13 18 60 8 15 17 20 60 Entries with a area = 210 a b c s area ----------------------------------- 17 25 28 35 210 20 21 29 35 210 12 35 37 42 210 17 28 39 42 210 7 65 68 70 210 3 148 149 150 210 FutureBasic text,,,,,70// Set width of tabs local fn gcd( a as long, b as long ) dim as long result if ( b != 0 ) result = fn gcd( b, a mod b) else result = abs(a) end if end fn = result begin globals dim as long triangleInfo( 600, 4 ) end globals local fn CalculateHeronianTriangles( numberToCheck as long ) as long dim as long c, b, a, result, count : count = 0 dim as double s, area for c = 1 to numberToCheck for b = 1 to c for a = 1 to b s = ( a + b + c ) / 2 area = s * ( s - a ) * ( s - b ) * ( s - c ) if area > 0 area = sqr( area ) if area = int( area ) result = fn gcd( b, c ) result = fn gcd( a, result ) if result == 1 count++ triangleInfo( count, 0 ) = a triangleInfo( count, 1 ) = b triangleInfo( count, 2 ) = c triangleInfo( count, 3 ) = 2 * s triangleInfo( count, 4 ) = area end if end if end if next next next end fn = count dim as long i, k, count count = fn CalculateHeronianTriangles( 200 ) print print "Number of triangles:"; count print print "---------------------------------------------" print "Side A", "Side B", "Side C", "Perimeter", "Area" print "---------------------------------------------" // Sort array dim as Boolean flips : flips = _true while ( flips = _true ) flips = _false for i = 1 to count - 1 if triangleInfo( i, 4 ) > triangleInfo( i + 1, 4 ) for k = 0 to 4 swap triangleInfo( i, k ), triangleInfo( i + 1, k ) next flips = _true end if next wend // Find first 10 heronian triangles for i = 1 to 10 print triangleInfo( i, 0 ), triangleInfo( i, 1 ), triangleInfo( i, 2 ), triangleInfo( i, 3 ), triangleInfo( i, 4 ) next print print "Triangles with an area of 210:" print // Search for triangle with area of 210 for i = 1 to count if triangleInfo( i, 4 ) == 210 print triangleInfo( i, 0 ), triangleInfo( i, 1 ), triangleInfo( i, 2 ), triangleInfo( i, 3 ), triangleInfo( i, 4 ) end if next HandleEvents Output: Number of triangles: 517 --------------------------------------------- Side A Side B Side C Perimeter Area --------------------------------------------- 3 4 5 12 6 5 5 6 16 12 5 5 8 18 12 4 13 15 32 24 5 12 13 30 30 9 10 17 36 36 3 25 26 54 36 7 15 20 42 42 10 13 13 36 60 8 15 17 40 60 Triangles with an area of 210: 17 25 28 70 210 20 21 29 70 210 12 35 37 84 210 17 28 39 84 210 7 65 68 140 210 3 148 149 300 210  Go package main import ( "fmt" "math" "sort" ) const ( n = 200 header = "\nSides P A" ) func gcd(a, b int) int { leftover := 1 var dividend, divisor int if (a > b) { dividend, divisor = a, b } else { dividend, divisor = b, a } for (leftover != 0) { leftover = dividend % divisor if (leftover > 0) { dividend, divisor = divisor, leftover } } return divisor } func is_heron(h float64) bool { return h > 0 && math.Mod(h, 1) == 0.0 } // by_area_perimeter implements sort.Interface for [][]int based on the area first and perimeter value type by_area_perimeter [][]int func (a by_area_perimeter) Len() int { return len(a) } func (a by_area_perimeter) Swap(i, j int) { a[i], a[j] = a[j], a[i] } func (a by_area_perimeter) Less(i, j int) bool { return a[i][4] < a[j][4] || a[i][4] == a[j][4] && a[i][3] < a[j][3] } func main() { var l [][]int for c := 1; c <= n; c++ { for b := 1; b <= c; b++ { for a := 1; a <= b; a++ { if (gcd(gcd(a, b), c) == 1) { p := a + b + c s := float64(p) / 2.0 area := math.Sqrt(s * (s - float64(a)) * (s - float64(b)) * (s - float64(c))) if (is_heron(area)) { l = append(l, []int{a, b, c, p, int(area)}) } } } } } fmt.Printf("Number of primitive Heronian triangles with sides up to %d: %d", n, len(l)) sort.Sort(by_area_perimeter(l)) fmt.Printf("\n\nFirst ten when ordered by increasing area, then perimeter:" + header) for i := 0; i < 10; i++ { fmt.Printf("\n%3d", l[i]) } a := 210 fmt.Printf("\n\nArea = %d%s", a, header) for _, it := range l { if (it[4] == a) { fmt.Printf("\n%3d", it) } } }  Output: Number of primitive Heronian triangles with sides up to 200: 517 First ten when ordered by increasing area, then perimeter: Sides P A [ 3 4 5 12 6] [ 5 5 6 16 12] [ 5 5 8 18 12] [ 4 13 15 32 24] [ 5 12 13 30 30] [ 9 10 17 36 36] [ 3 25 26 54 36] [ 7 15 20 42 42] [ 10 13 13 36 60] [ 8 15 17 40 60] Area = 210 Sides P A [ 17 25 28 70 210] [ 20 21 29 70 210] [ 12 35 37 84 210] [ 17 28 39 84 210] [ 7 65 68 140 210] [ 3 148 149 300 210] Haskell import qualified Data.List as L import Data.Maybe import Data.Ord import Text.Printf -- Determine if a number n is a perfect square and return its square root if so. -- This is used instead of sqrt to avoid fixed sized floating point numbers. perfectSqrt :: Integral a => a -> Maybe a perfectSqrt n | n == 1 = Just 1 | n < 4 = Nothing | otherwise = let search low high = let guess = (low + high) div 2 square = guess ^ 2 next | square == n = Just guess | low == guess = Nothing | square < n = search guess high | otherwise = search low guess in next in search 0 n -- Determine the area of a Heronian triangle if it is one. heronTri :: Integral a => a -> a -> a -> Maybe a heronTri a b c = let -- Rewrite Heron's formula to factor out the term 16 under the root. areaSq16 = (a + b + c) * (b + c - a) * (a + c - b) * (a + b - c) (areaSq, r) = areaSq16 divMod 16 in if r == 0 then perfectSqrt areaSq else Nothing isPrimitive :: Integral a => a -> a -> a -> a isPrimitive a b c = gcd a (gcd b c) third (_, _, x, _, _) = x fourth (_, _, _, x, _) = x fifth (_, _, _, _, x) = x orders :: Ord b => [(a -> b)] -> a -> a -> Ordering orders [f] a b = comparing f a b orders (f:fx) a b = case comparing f a b of EQ -> orders fx a b n -> n main :: IO () main = do let range = [1 .. 200] tris :: [(Integer, Integer, Integer, Integer, Integer)] tris = L.sortBy (orders [fifth, fourth, third])$ map (\(a, b, c, d, e) -> (a, b, c, d, fromJust e))
$filter (isJust . fifth) [(a, b, c, a + b + c, heronTri a b c) | a <- range, b <- range, c <- range , a <= b, b <= c, isPrimitive a b c == 1] printTri (a, b, c, d, e) = printf "%3d %3d %3d %9d %4d\n" a b c d e printf "Heronian triangles found: %d\n\n"$ length tris
putStrLn "   Sides    Perimeter Area"
mapM_ printTri $take 10 tris putStrLn "" mapM_ printTri$ filter ((== 210) . fifth) tris

Output:
Heronian triangles found: 517

Sides    Perimeter Area
3   4   5        12    6
5   5   6        16   12
5   5   8        18   12
4  13  15        32   24
5  12  13        30   30
9  10  17        36   36
3  25  26        54   36
7  15  20        42   42
10  13  13        36   60
8  15  17        40   60

17  25  28        70  210
20  21  29        70  210
12  35  37        84  210
17  28  39        84  210
7  65  68       140  210
3 148 149       300  210

J

Hero's formula Implementation

a=: 0&{"1
b=: 1&{"1
c=: 2&{"1
s=: (a+b+c) % 2:
area=: 2 %: s*(s-a)*(s-b)*(s-c)                   NB. Hero's formula
perim=: +/"1
isPrimHero=: (0&~: * (= <.@:+))@area * 1 = a +. b +. c


We exclude triangles with zero area, triangles with complex area, non-integer area, and triangles whose sides share a common integer multiple.

Alternative Implementation

The implementation above uses the symbols as given in the formula at the top of the page, making it easier to follow along as well as spot any errors. That formula distinguishes between the individual sides of the triangles but J could easily treat these sides as a single entity or array. The implementation below uses this "typical J" approach:

perim=: +/"1
s=: -:@:perim
area=: [: %: s * [: */"1 s - ]                    NB. Hero's formula
isNonZeroInt=: 0&~: *. (= <.@:+)
isPrimHero=: isNonZeroInt@area *. 1 = +./&.|:


Required examples

   Tri=:(1-i.3)+"1]3 comb 202                     NB. distinct triangles with sides <= 200
HeroTri=: (#~ isPrimHero) Tri                  NB. all primitive Heronian triangles with sides <= 200

# HeroTri                                      NB. count triangles found
517

HeroTri=: (/: area ,. perim ,. ]) HeroTri      NB. sort by area, perimeter & sides

(,. _ ,. perim ,. area) 10 {. HeroTri          NB. tabulate sides, perimeter & area for top 10 triangles
3  4  5 _ 12  6
5  5  6 _ 16 12
5  5  8 _ 18 12
4 13 15 _ 32 24
5 12 13 _ 30 30
9 10 17 _ 36 36
3 25 26 _ 54 36
7 15 20 _ 42 42
10 13 13 _ 36 60
8 15 17 _ 40 60

(,. _ ,. perim ,. area) (#~ 210 = area) HeroTri NB. tablulate sides, perimeter & area for triangles with area = 210
17  25  28 _  70 210
20  21  29 _  70 210
12  35  37 _  84 210
17  28  39 _  84 210
7  65  68 _ 140 210
3 148 149 _ 300 210


Java

import java.util.ArrayList;

public class Heron {
public static void main(String[] args) {
ArrayList<int[]> list = new ArrayList<>();

for (int c = 1; c <= 200; c++) {
for (int b = 1; b <= c; b++) {
for (int a = 1; a <= b; a++) {

if (gcd(gcd(a, b), c) == 1 && isHeron(heronArea(a, b, c))){
int area = (int) heronArea(a, b, c);
list.add(new int[]{a, b, c, a + b + c, area});
}
}
}
}
sort(list);

System.out.printf("Number of primitive Heronian triangles with sides up "
+ "to 200: %d\n\nFirst ten when ordered by increasing area, then"
+ " perimeter:\nSides       Perimeter   Area", list.size());

for (int i = 0; i < 10; i++) {
System.out.printf("\n%d x %d x %d   %d      %d",
list.get(i)[0], list.get(i)[1], list.get(i)[2],
list.get(i)[3], list.get(i)[4]);
}

System.out.printf("\n\nArea = 210\nSides        Perimeter   Area");
for (int i = 0; i < list.size(); i++) {
if (list.get(i)[4] == 210)
System.out.printf("\n%d x %d x %d   %d      %d",
list.get(i)[0], list.get(i)[1], list.get(i)[2],
list.get(i)[3], list.get(i)[4]);
}
}

public static double heronArea(int a, int b, int c) {
double s = (a + b + c) / 2f;
return Math.sqrt(s * (s - a) * (s - b) * (s - c));
}

public static boolean isHeron(double h) {
return h % 1 == 0 && h > 0;
}

public static int gcd(int a, int b) {
int leftover = 1, dividend = a > b ? a : b, divisor = a > b ? b : a;
while (leftover != 0) {
leftover = dividend % divisor;
if (leftover > 0) {
dividend = divisor;
divisor = leftover;
}
}
return divisor;
}

public static void sort(ArrayList<int[]> list) {
boolean swapped = true;
int[] temp;
while (swapped) {
swapped = false;
for (int i = 1; i < list.size(); i++) {
if (list.get(i)[4] < list.get(i - 1)[4] ||
list.get(i)[4] == list.get(i - 1)[4] &&
list.get(i)[3] < list.get(i - 1)[3]) {
temp = list.get(i);
list.set(i, list.get(i - 1));
list.set(i - 1, temp);
swapped = true;
}
}
}
}
}

Output:
Number of primitive Heronian triangles with sides up to 200: 517

First ten when ordered by increasing area, then perimeter:
Sides		Perimeter	Area
3 x 4 x 5	12		6
5 x 5 x 6	16		12
5 x 5 x 8	18		12
4 x 13 x 15	32		24
5 x 12 x 13	30		30
9 x 10 x 17	36		36
3 x 25 x 26	54		36
7 x 15 x 20	42		42
10 x 13 x 13	36		60
8 x 15 x 17	40		60

Area = 210
Sides		Perimeter	Area
17 x 25 x 28	70		210
20 x 21 x 29	70		210
12 x 35 x 37	84		210
17 x 28 x 39	84		210
7 x 65 x 68	140		210
3 x 148 x 149	300		210

JavaScript

Imperative

window.onload = function(){
var list = [];
var j = 0;
for(var c = 1; c <= 200; c++)
for(var b = 1; b <= c; b++)
for(var a = 1; a <= b; a++)
if(gcd(gcd(a, b), c) == 1 && isHeron(heronArea(a, b, c)))
list[j++] = new Array(a, b, c, a + b + c, heronArea(a, b, c));
sort(list);
document.write("<h2>Primitive Heronian triangles with sides up to 200: " + list.length + "</h2><h3>First ten when ordered by increasing area, then perimeter:</h3><table><tr><th>Sides</th><th>Perimeter</th><th>Area</th><tr>");
for(var i = 0; i < 10; i++)
document.write("<tr><td>" + list[i][0] + " x " + list[i][1] + " x " + list[i][2] + "</td><td>" + list[i][3] + "</td><td>" + list[i][4] + "</td></tr>");
document.write("</table><h3>Area = 210</h3><table><tr><th>Sides</th><th>Perimeter</th><th>Area</th><tr>");
for(var i = 0; i < list.length; i++)
if(list[i][4] == 210)
document.write("<tr><td>" + list[i][0] + " x " + list[i][1] + " x " + list[i][2] + "</td><td>" + list[i][3] + "</td><td>" + list[i][4] + "</td></tr>");
function heronArea(a, b, c){
var s = (a + b + c)/ 2;
return Math.sqrt(s *(s -a)*(s - b)*(s - c));
}
function isHeron(h){
return h % 1 == 0 && h > 0;
}
function gcd(a, b){
var leftover = 1, dividend = a > b ? a : b, divisor = a > b ? b : a;
while(leftover != 0){
leftover = dividend % divisor;
if(leftover > 0){
dividend = divisor;
divisor = leftover;
}
}
return divisor;
}
function sort(list){
var swapped = true;
var temp = [];
while(swapped){
swapped = false;
for(var i = 1; i < list.length; i++){
if(list[i][4] < list[i - 1][4] || list[i][4] == list[i - 1][4] && list[i][3] < list[i - 1][3]){
temp = list[i];
list[i] = list[i - 1];
list[i - 1] = temp;
swapped = true;
}
}
}
}
}

Output:
Primitive Heronian triangles with sides up to 200: 517

First ten when ordered by increasing area, then perimeter:
Sides	Perimeter	Area
3 x 4 x 5	12	6
5 x 5 x 6	16	12
5 x 5 x 8	18	12
4 x 13 x 15	32	24
5 x 12 x 13	30	30
9 x 10 x 17	36	36
3 x 25 x 26	54	36
7 x 15 x 20	42	42
10 x 13 x 13	36	60
8 x 15 x 17	40	60

Area = 210
Sides	Perimeter	Area
17 x 25 x 28	70	210
20 x 21 x 29	70	210
12 x 35 x 37	84	210
17 x 28 x 39	84	210
7 x 65 x 68	140	210
3 x 148 x 149	300	210

Functional (ES5)

Using the list monad pattern to define a filtered cartesian product:

- Monadic bind/chain for lists is concat map.
- Return/inject for lists is λx -> [x]
- Monadic fail for lists is simply λx -> [].

List comprehension syntax is convenient and concise, but efficient use of it may be helped by a clearer understanding of the formally equivalent – but slightly more flexible – list monad pattern. See, for example List comprehension at wiki.haskell.org. (Haskell list comprehensions are themselves implemented in terms of concat map). ES6 JavaScript introduces syntactic sugar for list comprehensions, but the list monad pattern can already be used in ES5 – indeed in any language which supports the use of higher-order functions.

(function (n) {

var chain = function (xs, f) {                  // Monadic bind/chain
return [].concat.apply([], xs.map(f));
},

hArea = function (x, y, z) {
var s = (x + y + z) / 2,
a = s * (s - x) * (s - y) * (s - z);
return a ? Math.sqrt(a) : 0;
},

gcd = function (m, n) { return n ? gcd(n, m % n) : m; },

rng = function (m, n) {
return Array.apply(null, Array(n - m + 1)).map(function (x, i) {
return m + i;
});
},

sum = function (a, x) { return a + x; };

// DEFINING THE SORTED SUB-SET IN TERMS OF A LIST MONAD

var lstHeron = chain( rng(1, n), function (x) {
return chain( rng(x, n), function (y) {
return chain( rng(y, n), function (z) {

return (
(x + y > z) &&
gcd(gcd(x, y), z) === 1 &&            // Primitive.
(function () {                        // Heronian.
var a = hArea(x, y, z);
return a && (a === parseInt(a, 10))
})()
) ? [[x, y, z]] : [];                   // Monadic inject or fail

})})}).sort(function (a, b) {
var dArea = hArea.apply(null, a) - hArea.apply(null, b);
if (dArea) return dArea;
else {
var dPerim = a.reduce(sum, 0) - b.reduce(sum, 0);
return dPerim ? dPerim : (a[2] - b[2]);
}
});

// OUPUT FORMATTED AS TWO WIKITABLES

var lstColumns = ['Sides Perimeter Area'.split(' ')],
fnData = function (lst) {
return [JSON.stringify(lst), lst.reduce(sum, 0), hArea.apply(null, lst)];
},
wikiTable = function (lstRows, blnHeaderRow, strStyle) {
return '{| class="wikitable" ' + (
strStyle ? 'style="' + strStyle + '"' : ''
) + lstRows.map(function (lstRow, iRow) {
var strDelim = ((blnHeaderRow && !iRow) ? '!' : '|');

return '\n|-\n' + strDelim + ' ' + lstRow.map(function (v) {
return typeof v === 'undefined' ? ' ' : v;
}).join(' ' + strDelim + strDelim + ' ');
}).join('') + '\n|}';
};

return 'Found: ' + lstHeron.length +
' primitive Heronian triangles with sides up to ' + n + '.\n\n' +
'(Showing first 10, sorted by increasing area, ' +
'perimeter, and longest side)\n\n' +
wikiTable(
lstColumns.concat(lstHeron.slice(0, 10).map(fnData)),
true
) + '\n\n' +
'All primitive Heronian triangles in this range where area = 210\n' +
'\n(also in order of increasing perimeter and longest side)\n\n' +
wikiTable(
lstColumns.concat(lstHeron.filter(function (x) {
return 210 === hArea.apply(null, x);
}).map(fnData)),
true
) + '\n\n';

})(200);

Output:

Found: 517 primitive Heronian triangles with sides up to 200.

(Showing first 10, sorted by increasing area, perimeter, and longest side)

Sides Perimeter Area
[3,4,5] 12 6
[5,5,6] 16 12
[5,5,8] 18 12
[4,13,15] 32 24
[5,12,13] 30 30
[9,10,17] 36 36
[3,25,26] 54 36
[7,15,20] 42 42
[10,13,13] 36 60
[8,15,17] 40 60

All primitive Heronian triangles in this range where area = 210

(also in order of increasing perimeter and longest side)

Sides Perimeter Area
[17,25,28] 70 210
[20,21,29] 70 210
[12,35,37] 84 210
[17,28,39] 84 210
[7,65,68] 140 210
[3,148,149] 300 210

jq

Works with: jq version 1.4
# input should be an array of the lengths of the sides
def hero:
(add/2) as $s | ($s*($s - .[0])*($s - .[1])*($s - .[2])) as$a2
| if $a2 > 0 then ($a2 | sqrt) else 0 end;

def is_heronian:
hero as $h |$h > 0 and ($h|floor) ==$h;

def gcd3(x; y; z):
# subfunction expects [a,b] as input
def rgcd:
if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[ ([x,y] | rgcd), z ] | rgcd;

def rjust(width): tostring |  " " * (width - length) + .;

[ range(1; maxside+1) as $c | range(1;$c+1) as $b | range(1;$b+1) as $a | if ($a + $b) >$c and gcd3($a;$b; $c) == 1 then [$a,$b,$c] | if is_heronian then . else empty end
else empty
end ]

# sort by increasing area, perimeter, then sides
| sort_by( [ hero, add, .[2] ] )
| "The number of primitive Heronian triangles with sides up to \(maxside): \(length)",
"The first ten when ordered by increasing area, then perimeter, then maximum sides:",
"      perimeter area",
(.[0:10][] | "\(rjust(11)) \(add | rjust(3)) \(hero | rjust(4))" ),
"All those with area 210, ordered as previously:",
"      perimeter area",
( .[] | select( hero == 210 ) | "\(rjust(11)) \(add|rjust(3)) \(hero|rjust(4))" ) ;

task(200)
Output:
$time jq -n -r -f heronian.jq The number of primitive Heronian triangles with sides up to 200: 517 The first ten when ordered by increasing area, then perimeter, then maximum sides: perimeter area [3,4,5] 12 6 [5,5,6] 16 12 [5,5,8] 18 12 [4,13,15] 32 24 [5,12,13] 30 30 [9,10,17] 36 36 [3,25,26] 54 36 [7,15,20] 42 42 [10,13,13] 36 60 [8,15,17] 40 60 All those with area 210, ordered as previously: perimeter area [17,25,28] 70 210 [20,21,29] 70 210 [12,35,37] 84 210 [17,28,39] 84 210 [7,65,68] 140 210 [3,148,149] 300 210  Julia The type IntegerTriangle stores a triangle's sides (a, b, c), perimeter (p) and area (σ) as integers. The function isprimheronian checks whether the a triangle of integer sides is a primitive Heronian triangle and is called prior to construction of an IntegerTriangle. Types and Functions type IntegerTriangle{T<:Integer} a::T b::T c::T p::T σ::T end function IntegerTriangle{T<:Integer}(a::T, b::T, c::T) p = a + b + c s = div(p, 2) σ = isqrt(s*(s-a)*(s-b)*(s-c)) (x, y, z) = sort([a, b, c]) IntegerTriangle(x, y, z, p, σ) end function isprimheronian{T<:Integer}(a::T, b::T, c::T) p = a + b + c iseven(p) || return false gcd(a, b, c) == 1 || return false s = div(p, 2) t = s*(s-a)*(s-b)*(s-c) 0 < t || return false σ = isqrt(t) σ^2 == t end  Main slim = 200 ht = IntegerTriangle[] for a in 1:slim, b in a:slim, c in b:slim isprimheronian(a, b, c) || continue push!(ht, IntegerTriangle(a, b, c)) end sort!(ht, by=x->(x.σ, x.p, x.c)) print("The number of primitive Hernonian triangles having sides ≤ ") println(slim, " is ", length(ht)) tlim = 10 tlim = min(tlim, length(ht)) println() println("Tabulating the first (by σ, p, c) ", tlim, " of these:") println(" a b c σ p") for t in ht[1:tlim] println(@sprintf "%6d %3d %3d %4d %4d" t.a t.b t.c t.σ t.p) end tlim = 210 println() println("Tabulating those having σ = ", tlim, ":") println(" a b c σ p") for t in ht t.σ == tlim || continue t.σ == tlim || break println(@sprintf "%6d %3d %3d %4d %4d" t.a t.b t.c t.σ t.p) end  Output: The number of primitive Hernonian triangles having sides ≤ 200 is 517 Tabulating the first (by σ, p, c) 10 of these: a b c σ p 3 4 5 6 12 5 5 6 12 16 5 5 8 12 18 4 13 15 24 32 5 12 13 30 30 9 10 17 36 36 3 25 26 36 54 7 15 20 42 42 10 13 13 60 36 8 15 17 60 40 Tabulating those having σ = 210: a b c σ p 17 25 28 210 70 20 21 29 210 70 12 35 37 210 84 17 28 39 210 84 7 65 68 210 140 3 148 149 210 300 Kotlin Translation of: Scala import java.util.ArrayList object Heron { private val n = 200 fun run() { val l = ArrayList<IntArray>() for (c in 1..n) for (b in 1..c) for (a in 1..b) if (gcd(gcd(a, b), c) == 1) { val p = a + b + c val s = p / 2.0 val area = Math.sqrt(s * (s - a) * (s - b) * (s - c)) if (isHeron(area)) l.add(intArrayOf(a, b, c, p, area.toInt())) } print("Number of primitive Heronian triangles with sides up to$n: " + l.size)

sort(l)
print("\n\nFirst ten when ordered by increasing area, then perimeter:" + header)
for (i in 0 until 10) {
print(format(l[i]))
}
val a = 210
print("\n\nArea = $a" + header) l.filter { it[4] == a }.forEach { print(format(it)) } } private fun gcd(a: Int, b: Int): Int { var leftover = 1 var dividend = if (a > b) a else b var divisor = if (a > b) b else a while (leftover != 0) { leftover = dividend % divisor if (leftover > 0) { dividend = divisor divisor = leftover } } return divisor } fun sort(l: MutableList<IntArray>) { var swapped = true while (swapped) { swapped = false for (i in 1 until l.size) if (l[i][4] < l[i - 1][4] || l[i][4] == l[i - 1][4] && l[i][3] < l[i - 1][3]) { val temp = l[i] l[i] = l[i - 1] l[i - 1] = temp swapped = true } } } private fun isHeron(h: Double) = h.rem(1) == 0.0 && h > 0 private val header = "\nSides Perimeter Area" private fun format(a: IntArray) = "\n%3d x %3d x %3d %5d %10d".format(a[0], a[1], a[2], a[3], a[4]) } fun main(args: Array<String>) = Heron.run()  Output: Number of primitive Heronian triangles with sides up to 200: 517 First ten when ordered by increasing area, then perimeter: Sides Perimeter Area 3 x 4 x 5 12 6 5 x 5 x 6 16 12 5 x 5 x 8 18 12 4 x 13 x 15 32 24 5 x 12 x 13 30 30 9 x 10 x 17 36 36 3 x 25 x 26 54 36 7 x 15 x 20 42 42 10 x 13 x 13 36 60 8 x 15 x 17 40 60 Area = 210 Sides Perimeter Area 17 x 25 x 28 70 210 20 x 21 x 29 70 210 12 x 35 x 37 84 210 17 x 28 x 39 84 210 7 x 65 x 68 140 210 3 x 148 x 149 300 210 Logtalk Implemented as a parametric object, the solution to making primitive Heronian triangles would look something like this: % In this example we assume that A<=B<=C. % Non-pedagogical code would verify and force this. :- object(triangle(_A_, _B_, _C_)). :- public([a/1, b/1, c/1, area/1, perimeter/1, primitive/0]). a(_A_). b(_B_). c(_C_). area(A) :- AB is _A_ + _B_, AB @> _C_, % you can't make a triangle if one side is half or longer the perimeter s(S), A is sqrt(S * (S - _A_) * (S - _B_) * (S - _C_)). perimeter(P) :- P is _A_ + _B_ + _C_. primitive :- heronian, gcd(1). % helper predicates heronian :- integer(_A_), integer(_B_), integer(_C_), area(A), A > 0.0, 0.0 is float_fractional_part(A). gcd(G) :- G is gcd(_A_, gcd(_B_, _C_)). s(S) :- perimeter(P), S is P / 2. :- end_object.  A quickly hacked-together test that produces the output for the task assignment would look something like this: :- object(test_triangle). :- uses(integer, [between/3]). :- uses(list, [length/2, member/2, sort/3, take/3]). :- uses(logtalk, [print_message(information, heronian, Message) as print(Message)]). :- public(start/0). start :- gather_primitive_heronians(Primitives), length(Primitives, L), print('There are ~w primitive Heronian triangles with sides under 200.~n'+[L]), sort(order_by(area), Primitives, AreaSorted), take(10, AreaSorted, Area10), print(@'The first ten found, ordered by area, are:\n'), display_each_element(Area10), sort(order_by(perimeter), Primitives, PerimeterSorted), take(10, PerimeterSorted, Perimeter10), print(@'The first ten found, ordered by perimeter, are:\n'), display_each_element(Perimeter10), findall( t(A, B, C, 210.0, Perimeter), member(t(A, B, C, 210.0, Perimeter), Primitives), Area210 ), print(@'The list of those with an area of 210 is:\n'), display_each_element(Area210). % localized helper predicates % display a single element in the provided format display_single_element(t(A,B,C,Area,Perimeter)) :- format(F), print(F+[A, B, C, Area, Perimeter]). % display each element in a list of elements, printing a header first display_each_element(L) :- print(@' A B C Area Perimeter'), print(@'=== === === ======= ========='), forall(member(T, L), display_single_element(T)), print(@'\n'). format('~|~ t~w~3+~ t~w~4+~ t~w~4+~ t~w~8+~ t~w~7+'). % collect all the primitive heronian triangles within the boundaries of the provided task gather_primitive_heronians(Primitives) :- findall( t(A, B, C, Area, Perimeter), ( between(3, 200, A), between(A, 200, B), between(B, 200, C), triangle(A, B, C)::primitive, triangle(A, B, C)::area(Area), triangle(A, B, C)::perimeter(Perimeter) ), Primitives ). order_by(_, =, T, T) :- !. order_by(area, <, t(_,_,_,Area1,_), t(_,_,_,Area2,_)) :- Area1 < Area2, !. order_by(area, >, t(_,_,_,Area1,_), t(_,_,_,Area2,_)) :- Area1 > Area2, !. order_by(perimeter, <, t(_,_,_,_,Perimeter1), t(_,_,_,_,Perimeter2)) :- Perimeter1 < Perimeter2, !. order_by(perimeter, >, t(_,_,_,_,Perimeter1), t(_,_,_,_,Perimeter2)) :- Perimeter1 > Perimeter2, !. order_by(_, <, t(A1,_,_,_,_), t(A2,_,_,_,_)) :- A1 < A2, !. order_by(_, <, t(_,B1,_,_,_), t(_,B2,_,_,_)) :- B1 < B2, !. order_by(_, <, t(_,_,C1,_,_), t(_,_,C2,_,_)) :- C1 < C2, !. order_by(_, >, _, _). :- end_object.  Output: ?- test_triangle::start. % There are 517 primitive Heronian triangles with sides under 200. % The first ten found, ordered by area, are: % A B C Area Perimeter % === === === ======= ========= % 3 4 5 6.0 12 % 5 5 6 12.0 16 % 5 5 8 12.0 18 % 4 13 15 24.0 32 % 5 12 13 30.0 30 % 3 25 26 36.0 54 % 9 10 17 36.0 36 % 7 15 20 42.0 42 % 6 25 29 60.0 60 % 8 15 17 60.0 40 % % The first ten found, ordered by perimeter, are: % A B C Area Perimeter % === === === ======= ========= % 3 4 5 6.0 12 % 5 5 6 12.0 16 % 5 5 8 12.0 18 % 5 12 13 30.0 30 % 4 13 15 24.0 32 % 9 10 17 36.0 36 % 10 13 13 60.0 36 % 8 15 17 60.0 40 % 7 15 20 42.0 42 % 13 14 15 84.0 42 % % The list of those with an area of 210 is: % A B C Area Perimeter % === === === ======= ========= % 3 148 149 210.0 300 % 7 65 68 210.0 140 % 12 35 37 210.0 84 % 17 25 28 210.0 70 % 17 28 39 210.0 84 % 20 21 29 210.0 70 % true.  Lua -- Returns the details of the Heronian Triangle with sides a, b, c or nil if it isn't one local function tryHt( a, b, c ) local result local s = ( a + b + c ) / 2; local areaSquared = s * ( s - a ) * ( s - b ) * ( s - c ); if areaSquared > 0 then -- a, b, c does form a triangle local area = math.sqrt( areaSquared ); if math.floor( area ) == area then -- the area is integral so the triangle is Heronian result = { a = a, b = b, c = c, perimeter = a + b + c, area = area } end end return result end -- Returns the GCD of a and b local function gcd( a, b ) return ( b == 0 and a ) or gcd( b, a % b ) end -- Prints the details of the Heronian triangle t local function htPrint( t ) print( string.format( "%4d %4d %4d %4d %4d", t.a, t.b, t.c, t.area, t.perimeter ) ) end -- Prints headings for the Heronian Triangle table local function htTitle() print( " a b c area perimeter" ); print( "---- ---- ---- ---- ---------" ) end -- Construct ht as a table of the Heronian Triangles with sides up to 200 local ht = {}; for c = 1, 200 do for b = 1, c do for a = 1, b do local t = gcd( gcd( a, b ), c ) == 1 and tryHt( a, b, c ); if t then ht[ #ht + 1 ] = t end end end end -- sort the table on ascending area, perimiter and max side length -- note we constructed the triangles with c as the longest side table.sort( ht, function( a, b ) return a.area < b.area or ( a.area == b.area and ( a.perimeter < b.perimeter or ( a.perimiter == b.perimiter and a.c < b.c ) ) ) end ); -- Display the triangles print( "There are " .. #ht .. " Heronian triangles with sides up to 200" ); htTitle(); for htPos = 1, 10 do htPrint( ht[ htPos ] ) end print( " ..." ); print( "Heronian triangles with area 210:" ); htTitle(); for htPos = 1, #ht do local t = ht[ htPos ]; if t.area == 210 then htPrint( t ) end end  Output: There are 517 Heronian triangles with sides up to 200 a b c area perimeter ---- ---- ---- ---- --------- 3 4 5 6 12 5 5 6 12 16 5 5 8 12 18 4 13 15 24 32 5 12 13 30 30 9 10 17 36 36 3 25 26 36 54 7 15 20 42 42 10 13 13 60 36 8 15 17 60 40 ... Heronian triangles with area 210: a b c area perimeter ---- ---- ---- ---- --------- 17 25 28 210 70 20 21 29 210 70 12 35 37 210 84 17 28 39 210 84 7 65 68 210 140 3 148 149 210 300  Mathematica / Wolfram Language ClearAll[Heron] Heron[a_, b_, c_] := With[{s = (a + b + c)/2}, Sqrt[s (s - a) (s - b) (s - c)]] PrintTemporary[Dynamic[{a, b, c}]]; results = Reap[ Do[ If[a < b + c \[And] b < c + a \[And] c < a + b, If[GCD[a, b, c] == 1, If[IntegerQ[Heron[a, b, c]], Sow[<|"Sides" -> {a, b, c}, "Area" -> Heron[a, b, c], "Perimeter" -> a + b + c, "MaximumSide" -> Max[a, b, c]|>] ] ] ] , {a, 1, 200}, {b, a, 200}, {c, b, 200} ] ][[2, 1]]; results = SortBy[results, {#["Area"] &, #["Perimeter"] &, #["MaximumSide"] &}]; results // Length Take[results, 10] // Dataset Select[results, #["Area"] == 210 &] // Dataset  Output: 517 Sides Area Perimeter MaximumSide {3,4,5} 6 12 5 {5,5,6} 12 16 6 {5,5,8} 12 18 8 {4,13,15} 24 32 15 {5,12,13} 30 30 13 {9,10,17} 36 36 17 {3,25,26} 36 54 26 {7,15,20} 42 42 20 {10,13,13} 60 36 13 {8,15,17} 60 40 17 Sides Area Perimeter MaximumSide {17,25,28} 210 70 28 {20,21,29} 210 70 29 {12,35,37} 210 84 37 {17,28,39} 210 84 39 {7,65,68} 210 140 68 {3,148,149} 210 300 149 Nim import std/[math, algorithm, lenientops, strformat, sequtils] type HeronianTriangle = tuple[a, b, c: int; p: int; area: int] # Functions with three operands. func max(a, b, c: int): int = max(a, max(b, c)) func gcd(a, b, c: int): int = gcd(a, gcd(b, c)) func cmp(x, y: HeronianTriangle): int = ## Compare two Heronian triangles. result = cmp(x.area, y.area) if result == 0: result = cmp(x.p, y.p) if result == 0: result = cmp(max(x.a, x.b, x.c), max(y.a, y.b, y.c)) func $(t: HeronianTriangle): string =
## Return the representation of a Heronian triangle.
fmt"{t.a:3d}, {t.b:3d}, {t.c:3d} {t.p:7d} {t.area:8d}"

func hero(a, b, c: int): float =
## Return the area of a triangle using Hero's formula.
let s = (a + b + c) / 2
result = sqrt(s * (s - a) * (s - b) * (s - c))

func isHeronianTriangle(x: float): bool = x > 0 and ceil(x) == x

const Header = "    Sides      Perimeter  Area\n-------------  ---------  ----"

var list: seq[HeronianTriangle]
const Max = 200

for c in 1..Max:
for b in 1..c:
for a in 1..b:
let area = hero(a, b, c)
if area.isHeronianTriangle and gcd(a, b, c) == 1:
let t: HeronianTriangle = (a, b, c, a + b + c, area.toInt)

list.sort(cmp)
echo "Number of Heronian triangles: ", list.len

echo "\nOrdered list of first ten Heronian triangles:"
for t in list[0 ..< 10]: echo t

echo "\nOrdered list of Heronian triangles with area 210:"
for t in list.filterIt(it.area == 210): echo t

Output:
Number of Heronian triangles: 517

Ordered list of first ten Heronian triangles:
Sides      Perimeter  Area
-------------  ---------  ----
3,   4,   5      12        6
5,   5,   6      16       12
5,   5,   8      18       12
4,  13,  15      32       24
5,  12,  13      30       30
9,  10,  17      36       36
3,  25,  26      54       36
7,  15,  20      42       42
10,  13,  13      36       60
8,  15,  17      40       60

Ordered list of Heronian triangles with area 210:
Sides      Perimeter  Area
-------------  ---------  ----
17,  25,  28      70      210
20,  21,  29      70      210
12,  35,  37      84      210
17,  28,  39      84      210
7,  65,  68     140      210
3, 148, 149     300      210

ooRexx

Derived from REXX with some changes

/*REXX pgm generates primitive Heronian triangles by side length & area.*/
Call time 'R'
Numeric Digits 12
Parse Arg mxs area list
If mxs ='' Then mxs =200
If area='' Then area=210
If list='' Then list=10
tx='primitive Heronian triangles'
Call heronian mxs            /* invoke sub with max SIDES.     */
Say nt tx 'found with side length up to' mxs "(inclusive)."
Call show '2'
Call show '3'
Say time('E') 'seconds elapsed'
Exit

heronian:
abc.=0  /* abc.ar.p.* contains 'a b c' for area ar and perimeter p */
nt=0                              /* number of triangles found     */
min.=''
max.=''
mem.=0
ln=length(mxs)
Do a=3 To mxs
Do b=a To mxs
ab=a+b
Do c=b To mxs
If hgcd(a,b,c)=1 Then Do    /* GCD=1                         */
ar=heron_area()
If pos('.',ar)=0 Then Do  /* is an integer                 */
nt=nt+1                 /* a primitive Heronian triangle.*/
Call minmax '0P',p
Call minmax '0A',a
per=ab+c
abc_ar=right(per,4) right(a,4) right(b,4) right(c,4),
right(ar,5)
Call mem abc_ar
End
End
End
End
End
/*
say 'min.p='min.0p
say 'max.p='max.0p
say 'min.a='min.0a
say 'max.a='max.0a
*/
Return nt

hgcd: Procedure
Parse Arg x
Do j=2 For 2
y=arg(j)
Do Until _==0
_=x//y
x=y
y=_
End
End
Return x

minmax:
Parse Arg which,x
If min.which='' Then Do
min.which=x
max.which=x
End
Else Do
min.which=min(min.which,x)
max.which=max(max.which,x)
End
--Say which min.which '-' max.which
Return

heron_area:
p=ab+c                           /* perimeter                      */
s=p/2
ar2=s*(s-a)*(s-b)*(s-c)          /* area**2                        */
If pos(right(ar2,1),'014569')=0 Then /* ar2 cannot be              */
Return '.'                         /* square of an integer*/
If ar2>0 Then
ar=sqrt(ar2)                   /* area                           */
Else
ar='.'
Return ar

show: Parse Arg which
Say ''
Select
When which='2' Then Do
Say 'Listing of the first' list tx":"
Do i=1 To list
Call ot i,mem.i
End
End
When which='3' Then Do
Say 'Listing of the' tx "with area=210"
j=0
Do i=1 To mem.0
Parse Var mem.i per a b c area
If area=210 Then Do
j=j+1
Call ot j,mem.i
End
End
End
End
Return

ot: Parse Arg k,mem
Parse Var mem per a b c area
Say right(k,9)'     area:'right(area,6)||,
'      perimeter:'right(per,4)'     sides:',
right(a,3) right(b,3) right(c,3)
Return

mem:
Parse Arg e
Do i=1 To mem.0
If mem.i>>e Then Leave
End
Do j=mem.0 to i By -1
j1=j+1
mem.j1=mem.j
End
mem.i=e
mem.0=mem.0+1
Return
/* for "Classic" REXX
sqrt: procedure; parse arg x;if x=0 then return 0;d=digits();numeric digits 11
numeric form;  parse value format(x,2,1,,0) 'E0' with g 'E' _ .;  g=g*.5'E'_%2
p=d+d%4+2; m.=11;  do j=0 while p>9; m.j=p; p=p%2+1; end;  do k=j+5 to 0 by -1
if m.k>11 then numeric digits m.k;g=.5*(g+x/g);end;numeric digits d;return g/1
*/
/* for ooRexx */
::requires rxmath library
::routine sqrt
Return rxCalcSqrt(arg(1),14)

Output:
517 primitive Heronian triangles found with side length up to 200 (inclusive).

Listing of the first 10 primitive Heronian triangles:
1     area:     6      perimeter:  12     sides:   3   4   5
2     area:    12      perimeter:  16     sides:   5   5   6
3     area:    12      perimeter:  18     sides:   5   5   8
4     area:    30      perimeter:  30     sides:   5  12  13
5     area:    24      perimeter:  32     sides:   4  13  15
6     area:    36      perimeter:  36     sides:   9  10  17
7     area:    60      perimeter:  36     sides:  10  13  13
8     area:    60      perimeter:  40     sides:   8  15  17
9     area:    42      perimeter:  42     sides:   7  15  20
10     area:    84      perimeter:  42     sides:  13  14  15

Listing of the primitive Heronian triangles with area=210
1     area:   210      perimeter:  70     sides:  17  25  28
2     area:   210      perimeter:  70     sides:  20  21  29
3     area:   210      perimeter:  84     sides:  12  35  37
4     area:   210      perimeter:  84     sides:  17  28  39
5     area:   210      perimeter: 140     sides:   7  65  68
6     area:   210      perimeter: 300     sides:   3 148 149
26.054000 seconds elapsed 

PARI/GP

Heron(v)=my([a,b,c]=v); (a+b+c)*(-a+b+c)*(a-b+c)*(a+b-c) \\ returns 16 times the squared area
is(a,b,c)=(a+b+c)%2==0 && gcd(a,gcd(b,c))==1 && issquare(Heron([a,b,c]))
v=List(); for(a=1,200,for(b=a+1,200,for(c=b+1,200, if(is(a,b,c),listput(v, [a,b,c])))));
v=Vec(v); #v
vecsort(v, (a,b)->Heron(a)-Heron(b))[1..10]
vecsort(v, (a,b)->vecsum(a)-vecsum(b))[1..10]
vecsort(v, 3)[1..10] \\ shortcut: order by third component
u=select(v->Heron(v)==705600, v);
vecsort(u, (a,b)->Heron(a)-Heron(b))
vecsort(u, (a,b)->vecsum(a)-vecsum(b))
vecsort(u, 3) \\ shortcut: order by third component
Output:
%1 = [[1, 2, 3], [1, 3, 4], [1, 4, 5], [1, 5, 6], [1, 6, 7], [1, 7, 8], [1, 8, 9], [1, 9, 10], [1, 10, 11], [1, 11, 12]]
%2 = [[1, 2, 3], [1, 3, 4], [1, 4, 5], [2, 3, 5], [1, 5, 6], [3, 4, 5], [1, 6, 7], [2, 5, 7], [3, 4, 7], [1, 7, 8]]
%3 = [[1, 2, 3], [1, 3, 4], [1, 4, 5], [2, 3, 5], [3, 4, 5], [1, 5, 6], [1, 6, 7], [2, 5, 7], [3, 4, 7], [1, 7, 8]]
%4 = [[3, 148, 149], [7, 65, 68], [12, 35, 37], [17, 25, 28], [17, 28, 39], [20, 21, 29]]
%5 = [[17, 25, 28], [20, 21, 29], [12, 35, 37], [17, 28, 39], [7, 65, 68], [3, 148, 149]]
%6 = [[17, 25, 28], [20, 21, 29], [12, 35, 37], [17, 28, 39], [7, 65, 68], [3, 148, 149]]

Pascal

Translation of: Lua
program heronianTriangles ( input, output );
type
(* record to hold details of a Heronian triangle *)
Heronian    = record a, b, c, area, perimeter : integer end;
refHeronian = ^Heronian;

var

ht             : array [ 1 .. 1000 ] of refHeronian;
htCount, htPos : integer;
a, b, c, i     : integer;
lower, upper   : integer;
k, h, t        : refHeronian;
swapped        : boolean;

(* returns the details of the Heronian Triangle with sides a, b, c or nil if it isn't one *)
function tryHt( a, b, c : integer ) : refHeronian;
var
s, areaSquared, area : real;
t                    : refHeronian;
begin
s           := ( a + b + c ) / 2;
areaSquared := s * ( s - a ) * ( s - b ) * ( s - c );
t           := nil;
if areaSquared > 0 then begin
(* a, b, c does form a triangle *)
area    := sqrt( areaSquared );
if trunc( area ) = area then begin
(* the area is integral so the triangle is Heronian *)
new(t);
t^.a := a; t^.b := b; t^.c := c; t^.area := trunc( area ); t^.perimeter := a + b + c
end
end;
tryHt := t
end (* tryHt *) ;

(* returns the GCD of a and b *)
function gcd( a, b : integer ) : integer;
begin
if b = 0 then gcd := a else gcd := gcd( b, a mod b )
end (* gcd *) ;

(* prints the details of the Heronian triangle t *)
procedure htPrint( t : refHeronian ) ; begin writeln( t^.a:4, t^.b:5, t^.c:5, t^.area:5, t^.perimeter:10 ) end;
(* prints headings for the Heronian Triangle table *)
procedure htTitle ; begin writeln( '   a    b    c area perimeter' ); writeln( '---- ---- ---- ---- ---------' ) end;

begin
(* construct ht as a table of the Heronian Triangles with sides up to 200 *)
htCount := 0;
for c := 1 to 200 do begin
for b := 1 to c do begin
for a := 1 to b do begin
if gcd( gcd( a, b ), c ) = 1 then begin
t := tryHt( a, b, c );
if t <> nil then begin
htCount       := htCount + 1;
ht[ htCount ] := t
end
end
end
end
end;

(* sort the table on ascending area, perimeter and max side length *)
(* note we constructed the triangles with c as the longest side *)
lower := 1;
upper := htCount;
repeat
upper   := upper - 1;
swapped := false;
for i := lower to upper do begin
h := ht[ i     ];
k := ht[ i + 1 ];
if ( k^.area < h^.area ) or (   ( k^.area =  h^.area )
and (  ( k^.perimeter <  h^.perimeter )
or (   ( k^.perimeter = h^.perimeter )
and ( k^.c <  h^.c )
)
)
)
then begin
ht[ i     ] := k;
ht[ i + 1 ] := h;
swapped     := true
end
end;
until not swapped;

(* display the triangles *)
writeln( 'There are ', htCount:1, ' Heronian triangles with sides up to 200' );
htTitle;
for htPos := 1 to 10 do htPrint( ht[ htPos ] );
writeln( ' ...' );
writeln( 'Heronian triangles with area 210:' );
htTitle;
for htPos := 1 to htCount do begin
t := ht[ htPos ];
if t^.area = 210 then htPrint( t )
end
end.

Output:
There are 517 Heronian triangles with sides up to 200
a    b    c area perimeter
---- ---- ---- ---- ---------
3    4    5    6        12
5    5    6   12        16
5    5    8   12        18
4   13   15   24        32
5   12   13   30        30
9   10   17   36        36
3   25   26   36        54
7   15   20   42        42
10   13   13   60        36
8   15   17   60        40
...
Heronian triangles with area 210:
a    b    c area perimeter
---- ---- ---- ---- ---------
17   25   28  210        70
20   21   29  210        70
12   35   37  210        84
17   28   39  210        84
7   65   68  210       140
3  148  149  210       300


Perl

Translation of: Raku
use strict;
use warnings;
use List::Util qw(max);

sub gcd { $_[1] == 0 ?$_[0] : gcd($_[1],$_[0] % $_[1]) } sub hero { my ($a, $b,$c) = @_[0,1,2];
my $s = ($a + $b +$c) / 2;
sqrt $s*($s - $a)*($s - $b)*($s - $c); } sub heronian_area { my$hero = hero my ($a,$b, $c) = @_[0,1,2]; sprintf("%.0f",$hero) eq $hero ?$hero : 0
}

sub primitive_heronian_area {
my ($a,$b, $c) = @_[0,1,2]; heronian_area($a, $b,$c) if 1 == gcd $a, gcd$b, $c; } sub show { print " Area Perimeter Sides\n"; for (@_) { my ($area, $perim,$c, $b,$a) = @$_; printf "%7d %9d %d×%d×%d\n",$area, $perim,$a, $b,$c;
}
}

sub main {
my $maxside = shift // 200; my$first = shift // 10;
my $witharea = shift // 210; my @h; for my$c (1 .. $maxside) { for my$b (1 .. $c) { for my$a ($c -$b + 1 .. $b) { if (my$area = primitive_heronian_area $a,$b, $c) { push @h, [$area, $a+$b+$c,$c, $b,$a];
}
}
}
}
@h = sort {
$a->[0] <=>$b->[0]
or
$a->[1] <=>$b->[1]
or
max(@$a[2,3,4]) <=> max(@$b[2,3,4])
} @h;
printf "Primitive Heronian triangles with sides up to %d: %d\n",
$maxside, scalar @h; print "First:\n"; show @h[0 ..$first - 1];
print "Area $witharea:\n"; show grep {$_->[0] == $witharea } @h; } &main();  Output: Primitive Heronian triangles with sides up to 200: 517 First: Area Perimeter Sides 6 12 3×4×5 12 16 5×5×6 12 18 5×5×8 24 32 4×13×15 30 30 5×12×13 36 36 9×10×17 36 54 3×25×26 42 42 7×15×20 60 36 10×13×13 60 40 8×15×17 Area 210: Area Perimeter Sides 210 70 17×25×28 210 70 20×21×29 210 84 12×35×37 210 84 17×28×39 210 140 7×65×68 210 300 3×148×149 Phix function heroArea(integer a, b, c) atom s = (a+b+c)/2 return sqrt(max(s*(s-a)*(s-b)*(s-c),0)) end function function hero(atom h) return remainder(h,1)=0 and h>0 end function sequence list = {} integer tries = 0 for a=1 to 200 do for b=1 to a do for c=1 to b do tries += 1 if gcd({a,b,c})=1 then atom hArea = heroArea(a,b,c) if hero(hArea) then list = append(list,{hArea,a+b+c,a,b,c}) end if end if end for end for end for list = sort(list) printf(1,"Primitive Heronian triangles with sides up to 200: %d (of %,d tested)\n\n",{length(list),tries}) printf(1,"First 10 ordered by area/perimeter/sides:\n") printf(1,"area perimeter sides\n") for i=1 to 10 do printf(1,"%4d %3d %dx%dx%d\n",list[i]) end for printf(1,"\narea = 210:\n") printf(1,"area perimeter sides\n") for i=1 to length(list) do if list[i][1]=210 then printf(1,"%4d %3d %dx%dx%d\n",list[i]) end if end for  Output: Primitive Heronian triangles with sides up to 200: 517 (of 1,353,400 tested) First 10 ordered by area/perimeter/sides: area perimeter sides 6 12 5x4x3 12 16 6x5x5 12 18 8x5x5 24 32 15x13x4 30 30 13x12x5 36 36 17x10x9 36 54 26x25x3 42 42 20x15x7 60 36 13x13x10 60 40 17x15x8 area = 210: area perimeter sides 210 70 28x25x17 210 70 29x21x20 210 84 37x35x12 210 84 39x28x17 210 140 68x65x7 210 300 149x148x3  PowerShell function Get-Gcd($a, $b){ if($a -ge $b){$dividend = $a$divisor = $b } else{$dividend = $b$divisor = $a }$leftover = 1
while($leftover -ne 0){$leftover = $dividend %$divisor
if($leftover -ne 0){$dividend = $divisor$divisor = $leftover } }$divisor
}
function Is-Heron($heronArea){$heronArea -gt 0 -and $heronArea % 1 -eq 0 } function Get-HeronArea($a, $b,$c){
$s = ($a + $b +$c) / 2
[math]::Sqrt($s * ($s - $a) * ($s - $b) * ($s - $c)) }$result = @()
foreach ($c in 1..200){ for($b = 1; $b -le$c; $b++){ for($a = 1; $a -le$b; $a++){ if((Get-Gcd$c (Get-Gcd $b$a)) -eq 1 -and (Is-Heron(Get-HeronArea $a$b $c))){$result += @(,@($a,$b, $c,($a + $b +$c), (Get-HeronArea $a$b $c))) } } } }$result = $result | sort-object @{Expression={$_[4]}}, @{Expression={$_[3]}}, @{Expression={$_[2]}}
"Primitive Heronian triangles with sides up to 200: $($result.length)nFirst ten when ordered by increasing area, then perimeter,then maximum sides:nSidesttttPerimetertArea"
for($i = 0;$i -lt 10; $i++){ "$($result[$i][0])t$($result[$i][1])t$($result[$i][2])ttt$($result[$i][3])ttt$($result[$i][4])"
}
"nArea = 210nSidesttttPerimetertArea"
foreach($i in$result){
if($i[4] -eq 210){ "$($i[0])t$($i[1])t$($i[2])ttt$($i[3])ttt$($i[4])" } }  Output: Primitive Heronian triangles with sides up to 200: 517 First ten when ordered by increasing area, then perimeter,then maximum sides: Sides Perimeter Area 3 4 5 12 6 5 5 6 16 12 5 5 8 18 12 4 13 15 32 24 5 12 13 30 30 9 10 17 36 36 3 25 26 54 36 7 15 20 42 42 10 13 13 36 60 8 15 17 40 60 Area = 210 Sides Perimeter Area 17 25 28 70 210 20 21 29 70 210 12 35 37 84 210 17 28 39 84 210 7 65 68 140 210 3 148 149 300 210  Python from __future__ import division, print_function from math import gcd, sqrt def hero(a, b, c): s = (a + b + c) / 2 a2 = s * (s - a) * (s - b) * (s - c) return sqrt(a2) if a2 > 0 else 0 def is_heronian(a, b, c): a = hero(a, b, c) return a > 0 and a.is_integer() def gcd3(x, y, z): return gcd(gcd(x, y), z) if __name__ == '__main__': MAXSIDE = 200 N = 1 + MAXSIDE h = [(x, y, z) for x in range(1, N) for y in range(x, N) for z in range(y, N) if (x + y > z) and 1 == gcd3(x, y, z) and is_heronian(x, y, z)] # By increasing area, perimeter, then sides h.sort(key=lambda x: (hero(*x), sum(x), x[::-1])) print( 'Primitive Heronian triangles with sides up to %i:' % MAXSIDE, len(h) ) print('\nFirst ten when ordered by increasing area, then perimeter,', 'then maximum sides:') print('\n'.join(' %14r perim: %3i area: %i' % (sides, sum(sides), hero(*sides)) for sides in h[:10])) print('\nAll with area 210 subject to the previous ordering:') print('\n'.join(' %14r perim: %3i area: %i' % (sides, sum(sides), hero(*sides)) for sides in h if hero(*sides) == 210))  Output: Primitive Heronian triangles with sides up to 200: 517 First ten when ordered by increasing area, then perimeter,then maximum sides: (3, 4, 5) perim: 12 area: 6 (5, 5, 6) perim: 16 area: 12 (5, 5, 8) perim: 18 area: 12 (4, 13, 15) perim: 32 area: 24 (5, 12, 13) perim: 30 area: 30 (9, 10, 17) perim: 36 area: 36 (3, 25, 26) perim: 54 area: 36 (7, 15, 20) perim: 42 area: 42 (10, 13, 13) perim: 36 area: 60 (8, 15, 17) perim: 40 area: 60 All with area 210 subject to the previous ordering: (17, 25, 28) perim: 70 area: 210 (20, 21, 29) perim: 70 area: 210 (12, 35, 37) perim: 84 area: 210 (17, 28, 39) perim: 84 area: 210 (7, 65, 68) perim: 140 area: 210 (3, 148, 149) perim: 300 area: 210 R Mostly adopted from Python implementation: area <- function(a, b, c) { s = (a + b + c) / 2 a2 = s*(s-a)*(s-b)*(s-c) if (a2>0) sqrt(a2) else 0 } is.heronian <- function(a, b, c) { h = area(a, b, c) h > 0 && 0==h%%1 } # borrowed from stackoverflow http://stackoverflow.com/questions/21502181/finding-the-gcd-without-looping-r gcd <- function(x,y) { r <- x%%y; ifelse(r, gcd(y, r), y) } gcd3 <- function(x, y, z) { gcd(gcd(x, y), z) } maxside = 200 r <- NULL for(c in 1:maxside){ for(b in 1:c){ for(a in 1:b){ if(1==gcd3(a, b, c) && is.heronian(a, b, c)) { r <- rbind(r,c(a=a, b=b, c=c, perimeter=a+b+c, area=area(a,b,c))) } } } } cat("There are ",nrow(r)," Heronian triangles up to a maximal side length of ",maxside,".\n", sep="") cat("Showing the first ten ordered first by perimeter, then by area:\n") print(head(r[order(x=r[,"perimeter"],y=r[,"area"]),],n=10))  Output: There are 517 Heronian triangles up to a maximal side length of 200. Showing the first ten ordered first by perimeter, then by area: a b c perimeter area [1,] 3 4 5 12 6 [2,] 5 5 6 16 12 [3,] 5 5 8 18 12 [4,] 5 12 13 30 30 [5,] 4 13 15 32 24 [6,] 9 10 17 36 36 [7,] 10 13 13 36 60 [8,] 8 15 17 40 60 [9,] 7 15 20 42 42 [10,] 13 14 15 42 84  Racket #lang at-exp racket (require data/order scribble/html) ;; Returns the area of a triangle iff the sides have gcd 1, and it is an ;; integer; #f otherwise (define (heronian?-area a b c) (and (= 1 (gcd a b c)) (let ([s (/ (+ a b c) 2)]) ; ** If s=\frac{a+b+c}{2} (and (integer? s) ; (s must be an integer for the area to b an integer) (let-values ([[q r] (integer-sqrt/remainder ; (faster than sqrt) ; ** Then the area is \sqrt{s(s-a)(s-b)(s-c)} (* s (- s a) (- s b) (- s c)))]) (and (zero? r) q)))))) ; (return only integer areas) (define (generate-heronian-triangles max-side) (for*/list ([c (in-range 1 (add1 max-side))] [b (in-range 1 (add1 c))] ; b<=c [a (in-range (add1 (- c b)) (add1 b))] ; ensures a<=b and c<a+b [area (in-value (heronian?-area a b c))] #:when area) ;; datum-order can sort this for the tables (c is the max side length) (list area (+ a b c) c (list a b c)))) ;; Order the triangles by first increasing area, then by increasing perimeter, ;; then by increasing maximum side lengths (define (tri-sort triangles) (sort triangles (λ(t1 t2) (eq? '< (datum-order t1 t2))))) (define (triangles->table triangles) (table (tr (map th '("#" sides perimeter area))) "\n" (for/list ([i (in-naturals 1)] [triangle (in-list triangles)]) (match-define (list area perimeter max-side sides) triangle) (tr (td i) (td (add-between sides ",")) (td perimeter) (td area) "\n")))) (module+ main (define ts (generate-heronian-triangles 200)) (output-xml @div{@p{number of primitive triangles found with perimeter @entity{le} 200 = @(length ts)} @; Show the first ten ordered triangles in a table of sides, perimeter, @; and area. @(triangles->table (take (tri-sort ts) 10)) @; Show a similar ordered table for those triangles with area = 210 @(triangles->table (tri-sort (filter (λ(t) (eq? 210 (car t))) ts))) }))  This program generates HTML, so the output is inline with the page, not in a <pre> block. Output: number of primitive triangles found with perimeter ≤ 200 = 517 #sidesperimeterarea 13,4,5126 25,5,61612 35,5,81812 44,13,153224 55,12,133030 69,10,173636 73,25,265436 87,15,204242 910,13,133660 108,15,174060 #sidesperimeterarea 117,25,2870210 220,21,2970210 312,35,3784210 417,28,3984210 57,65,68140210 63,148,149300210 Raku (formerly Perl 6) Works with: Rakudo version 2018.09 sub hero($a, $b,$c) {
my $s = ($a + $b +$c) / 2;
($s * ($s - $a) * ($s - $b) * ($s - $c)).sqrt; } sub heronian-area($a, $b,$c) {
$_ when Int given hero($a, $b,$c).narrow;
}

sub primitive-heronian-area($a,$b, $c) { heronian-area$a, $b,$c
if 1 == [gcd] $a,$b, $c; } sub show(@measures) { say " Area Perimeter Sides"; for @measures -> [$area, $perim,$c, $b,$a] {
printf "%6d %6d %12s\n", $area,$perim, "$a×$b×$c"; } } sub MAIN ($maxside = 200, $first = 10,$witharea = 210) {
my @hh[1000];
my atomicint $i; (1 ..$maxside).race(:12batch).map: -> $c { for 1 ..$c -> $b { for$c - $b + 1 ..$b -> $a { if primitive-heronian-area($a,$b,$c) -> $area { @hh[$i⚛++] = [$area,$a+$b+$c, $c,$b, $a]; } } } } my @h = (@hh.grep: so *).sort; say "Primitive Heronian triangles with sides up to$maxside: ", +@h;

say "\nFirst $first:"; show @h[^$first];

say "\nArea $witharea:"; show @h.grep: *[0] ==$witharea;
}

Output:
Primitive Heronian triangles with sides up to 200: 517

First 10:
Area Perimeter   Sides
6     12        3×4×5
12     16        5×5×6
12     18        5×5×8
24     32      4×13×15
30     30      5×12×13
36     36      9×10×17
36     54      3×25×26
42     42      7×15×20
60     36     10×13×13
60     40      8×15×17

Area 210:
Area Perimeter   Sides
210     70     17×25×28
210     70     20×21×29
210     84     12×35×37
210     84     17×28×39
210    140      7×65×68
210    300    3×148×149

REXX

using iSQRT

This REXX version makes use of these facts:

•   if   A   is even,   then   B   and   C   must be odd.
•   if   B   is even,   then   C                 must be odd.
•   if   A   and   B   are odd,   then   C   must be even.
•   with the 1st three truisms, then:
•   C   can be incremented by   2.
•   the area is always even.

Programming notes:

The   hGCD   subroutine is a specialized version of a GCD routine in that:

•   it doesn't check for non-positive integers
•   it expects exactly three arguments

Also, a fair amount of code was added to optimize the speed   (at the expense of program simplicity).

By thoughtful ordering of the elimination checks, and also the use of an   integer version   of a   SQRT
subroutine,   the execution time was greatly reduced   (by a factor of eight).

Note that the   hIsqrt   (heronian Integer sqare root)   subroutine doesn't use floating point.
[hIsqrt   is a modified/simplified version of an   Isqrt   function.]

This REXX version doesn't need to explicitly sort the triangles as they are listed in the proper order.

/*REXX program generates & displays primitive Heronian triangles by side length and area*/
parse arg  N  first  area  .                     /*obtain optional arguments from the CL*/
if     N==''  |     N==","  then     N= 200      /*Not specified?  Then use the default.*/
if first==''  | first==","  then first=  10      /* "      "         "   "   "     "    */
if  area==''  |  area==","  then  area= 210      /* "      "         "   "   "     "    */
numeric digits 99                                /*ensure 'nuff dec. digs to calc. N**5.*/
numeric digits max(9, 1 + length(N**5) )         /*minimize decimal digits for REXX pgm.*/
call Heron;       HT= 'Heronian triangles'       /*invoke the  Heron  subroutine.       */
say  #          ' primitive'  HT  "found with sides up to "   N  ' (inclusive).'
call show     , 'Listing of the first '      first      ' primitive'      HT":"
call show area, 'Listing of the (above) found primitive'   HT   "with an area of "    area
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Heron: @.= 0;  minP= 9e9;  maxP= 0;  maxA= 0;  minA= 9e9;  Ln= length(N)    /*        __*/
#= 0;  #.= 0;  #.2= 1;  #.3= 1;  #.7= 1;  #.8= 1       /*digits ¬good √  */
do a=3  to N                                   /*start at a minimum side length of 3. */
Aeven= a//2==0                        /*if  A  is even,  B and C must be odd.*/
do b=a+Aeven  to N  by 1+Aeven;   ab= a + b  /*AB: a shortcut for the sum of A & B. */
if b//2==0  then                bump= 1      /*Is  B  even?       Then  C  is odd.  */
else if Aeven  then bump= 0      /*Is  A  even?         "   "   "  "    */
else bump= 1      /*A and B  are both odd,  biz as usual.*/
do c=b+bump  to N  by 2;   s= (ab + c) % 2 /*calculate triangle's perimeter:   S. */
_= s*(s-a)*(s-b)*(s-c); if _<=0   then iterate /*is  _  not positive?      Skip it*/
parse var _ '' -1 z   ; if #.z    then iterate /*Last digit not square?    Skip it*/
ar= hIsqrt(_);       if ar*ar\==_ then iterate /*Is area not an integer?   Skip it*/
if hGCD(a, b, c) \== 1            then iterate /*GCD of sides not equal 1? Skip it*/
#= # + 1;                p= ab + c             /*primitive Heronian triangle.     */
minP= min( p, minP);     maxP= max( p, maxP);        Lp= length(maxP)
minA= min(ar, minA);     maxA= max(ar, maxA);        La= length(maxA)
_=@.ar.p.0 + 1                                 /*bump Heronian triangle counter.  */
@.ar.p.0= _;  @.ar.p._= right(a, Ln)   right(b, Ln)   right(c, Ln)       /*unique.*/
end   /*c*/                                    /* [↑]  keep each unique perimeter#*/
end     /*b*/
end       /*a*/;             return #              /*return # of Heronian triangles.  */
/*──────────────────────────────────────────────────────────────────────────────────────*/
hGCD: x=a;   do j=2  for 2;    y= arg(j);       do until y==0; parse value x//y y with y x
end   /*until*/
end   /*j*/;      return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
hIsqrt: procedure; parse arg x;  q= 1;  r= 0;                  do  while q<=x;    q= q * 4
end   /*while q<=x*/
do  while q>1; q=q%4; _= x-r-q; r= r%2; if _>=0  then parse value _ r+q with x r
end   /*while q>1*/;          return r
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: m=0;  say;  say;   parse arg ae;     say arg(2);         if ae\==''  then first= 9e9
say;  $=left('',9);$a= $"area:";$p= $'perimeter:';$s= $"sides:" /*literals*/ do i=minA to maxA; if ae\=='' & i\==ae then iterate /*= area? */ do j=minP to maxP until m>=first /*only display the FIRST entries.*/ do k=1 for @.i.j.0; m= m + 1 /*display each perimeter entry. */ say right(m, 9)$a    right(i, La)    $p right(j, Lp)$s    @.i.j.k
end   /*k*/
end     /*j*/                           /* [↑]  use the known perimeters. */
end       /*i*/;            return        /* [↑]  show any found triangles. */

output   when using the default inputs:
517  primitive Heronian triangles found with sides up to  200  (inclusive).

Listing of the first  10  primitive Heronian triangles:

1          area:     6          perimeter:  12          sides:   3   4   5
2          area:    12          perimeter:  16          sides:   5   5   6
3          area:    12          perimeter:  18          sides:   5   5   8
4          area:    24          perimeter:  32          sides:   4  13  15
5          area:    30          perimeter:  30          sides:   5  12  13
6          area:    36          perimeter:  36          sides:   9  10  17
7          area:    36          perimeter:  54          sides:   3  25  26
8          area:    42          perimeter:  42          sides:   7  15  20
9          area:    60          perimeter:  36          sides:  10  13  13
10          area:    60          perimeter:  40          sides:   8  15  17

Listing of the (above) found primitive Heronian triangles with an area of  210

1          area:   210          perimeter:  70          sides:  17  25  28
2          area:   210          perimeter:  70          sides:  20  21  29
3          area:   210          perimeter:  84          sides:  12  35  37
4          area:   210          perimeter:  84          sides:  17  28  39
5          area:   210          perimeter: 140          sides:   7  65  68
6          area:   210          perimeter: 300          sides:   3 148 149


using SQRT table

This REXX version makes use of a precalculated table of squares of some integers   (which are used to find square roots very quickly).

It is about eight times faster than the 1st REXX version.

/*REXX program generates & displays primitive Heronian triangles by side length and area*/
parse arg  N  first  area  .                     /*obtain optional arguments from the CL*/
if     N==''  |     N==","  then     N= 200      /*Not specified?  Then use the default.*/
if first==''  | first==","  then first=  10      /* "      "         "   "   "     "    */
if  area==''  |  area==","  then  area= 210      /* "      "         "   "   "     "    */
numeric digits 99; numeric digits max(9, 1+length(N**5))  /*ensure 'nuff decimal digits.*/
call Heron;       HT= 'Heronian triangles'       /*invoke the  Heron  subroutine.       */
say  #          ' primitive'    HT    "found with sides up to "     N      ' (inclusive).'
call show     , 'Listing of the first '       first        ' primitive'            HT":"
call show area, 'Listing of the (above) found primitive'   HT   "with an area of "    area
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Heron: @.= 0;   #= 0;   !.= .;   minP= 9e9;  maxA= 0;  maxP= 0;  minA= 9e9;  Ln= length(N)
do i=1  for N**2%2;    _= i*i;      !._= i               /*     __   */
end   /*i*/                   /* [↑]  pre─calculate some fast  √     */
do a=3  to N                                   /*start at a minimum side length of 3. */
Aeven= a//2==0                        /*if  A  is even,  B and C must be odd.*/
do b=a+Aeven  to N  by 1+Aeven;   ab= a + b  /*AB: a shortcut for the sum of A & B. */
if b//2==0  then                bump= 1      /*Is  B  even?       Then  C  is odd.  */
else if Aeven  then bump= 0      /*Is  A  even?         "   "   "  "    */
else bump= 1      /*A and B  are both odd,  biz as usual.*/
do c=b+bump  to N  by 2;   s= (ab + c) % 2 /*calculate triangle's perimeter:   S. */
_= s*(s-a)*(s-b)*(s-c);  if !._==.     then iterate  /*Is  _  not a square?  Skip.*/
if hGCD(a,b,c) \== 1                   then iterate  /*GCD of sides not 1?   Skip.*/
#= # + 1;     p= ab + c;   ar= !._                   /*primitive Heronian triangle*/
minP= min( p, minP);     maxP= max( p, maxP);       Lp= length(maxP)
minA= min(ar, minA);     maxA= max(ar, maxA);       La= length(maxA);         @.ar=
_= @.ar.p.0  +  1                                    /*bump the triangle counter. */
@.ar.p.0= _;    @.ar.p._= right(a, Ln)    right(b, Ln)    right(c, Ln)    /*unique*/
end   /*c*/                                /* [↑]  keep each unique perimeter #.  */
end     /*b*/
end       /*a*/;    return #                   /*return number of Heronian triangles. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
hGCD: x=a;  do j=2  for 2;   y= arg(j);         do until y==0; parse value x//y y with y x
end   /*until*/
end   /*j*/;                return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: m=0;  say;  say;   parse arg ae;     say arg(2);         if ae\==''  then first= 9e9
say;  $=left('',9);$a= $"area:";$p= $'perimeter:';$s= $"sides:" /*literals*/ do i=minA to maxA; if ae\=='' & i\==ae then iterate /*= area? */ do j=minP to maxP until m>=first /*only display the FIRST entries.*/ do k=1 for @.i.j.0; m= m + 1 /*display each perimeter entry. */ say right(m, 9)$a    right(i, La)    $p right(j, Lp)$s    @.i.j.k
end   /*k*/
end     /*j*/                           /* [↑]  use the known perimeters. */
end       /*i*/;            return        /* [↑]  show any found triangles. */

output   is identical to the 1st REXX version.

Ring

# Project : Heronian triangles

see "Heronian triangles with sides up to 200" + nl
see "Sides               Perimeter       Area" + nl
for n = 1 to 200
for m = n to 200
for p = m to 200
s = (n + m + p) / 2
w = sqrt(s * (s - n) * (s - m) * (s - p))
bool = (gcd(n, m) = 1 or gcd(n, m) = n) and (gcd(n, p) = 1 or gcd(n, p) = n) and (gcd(m, p) = 1 or gcd(m, p) = m)
if w = floor(w) and w > 0 and bool
see "{" + n + ", " + m + ", " + p + "}" + "              " + s*2 + "              " + w + nl
ok
next
next
next
see nl

see "Heronian triangles with area 210:" + nl
see "Sides               Perimeter       Area" + nl
for n = 1 to 150
for m = n to 150
for p = m to 150
s = (n + m + p) / 2
w = sqrt(s * (s - n) * (s - m) * (s - p))
bool = (gcd(n, m) = 1 or gcd(n, m) = n) and (gcd(n, p) = 1 or gcd(n, p) = n) and (gcd(m, p) = 1 or gcd(m, p) = m)
if w = 210 and bool
see "{" + n + ", " + m + ", " + p + "}" + "              " + s*2 + "              " + w + nl
ok
next
next
next

func gcd(gcd, b)
while b
c   = gcd
gcd = b
b   = c % b
end
return gcd

Output:

Heronian triangles with sides up to 200
Sides               Perimeter       Area
{3, 4, 5}              12            6
{3, 25, 26}          54            36
{4, 13, 15}          32            24
{5, 5, 6}              16            12
{5, 5, 8}              18            12
{5, 12, 13}          30            30
{7, 15, 20 }         42            42
{8, 15, 17}          40            60
{9, 10, 17}          36            36
{10, 13, 13}         36            60
{13, 13, 24}         50            60

Heronian triangles with area 210:
Sides               Perimeter       Area
{3, 148, 149}         300          210
{7, 65, 68}            140           210
{12, 35, 37}            84           210
{17, 25, 28}            70           210
{17, 28, 39}            84           210
{20, 21, 29}            70           210


RPL

We use here the →V3 andSORT instructions, available for HP-48G or newer models only. GCD  is not a built-in instruction, but it is a question of a few words:

 ≪ WHILE DUP REPEAT SWAP OVER MOD END DROP ABS ≫ 'GCD' STO

Translation of: FreeBASIC
Works with: Halcyon Calc version 4.2.8
RPL code Comment
≪
3 DUPN + + 2 / → a b c s
≪ s DUP a - * s b - * s c - *
≫ ≫ ‘SURF2’ STO

≪
IF SURF2 DUP 0 > THEN √ FP NOT ELSE DROP 0 END
≫ ‘HERO?’ STO

≪ → n
≪ { } 1 n FOR x
x n FOR y
y x 2 MOD + x y + 1 - n MIN FOR z
IF x y z GCD GCD THEN
IF x y z HERO? THEN x y z →V3 +
END END
2 STEP NEXT NEXT

SURF2 ( a b c → A² )
s = (a+b+c)/2
A² = s(s-a)(s-b)(s-c)
return A²

HERO? ( a b c → boolean )
return true if A > 0 and √A is an integer

TASK2 ( n → { [Heronians] )
for x = 1 to n
for y = x to n
for z = y+u to min(x+y-1,n)  // u ensures x+y+z is even
if gcd(x,y,z) == 1
if x y z is Heronian then append to list

z += 2 to keep x+y+z even



The rest of the code, which is devoted to printing the requested tables, is boring and the result is awful: native RPL works on machines with a 22-character screen.

Input:
≪ → a b c
≪ c →STR WHILE DUP SIZE 4 < REPEAT " " SWAP + END
a b c + + →STR SWAP + WHILE DUP SIZE 8 < REPEAT " " SWAP + END
a b c SURF √ →STR SWAP + WHILE DUP SIZE 12 < REPEAT " " SWAP + END
" (" + a →STR + " " + b →STR + " " + c →STR + ")" +
≫ ≫ 'PRTRI' STO
≪ { } 1 H SIZE FOR j H j GET ARRY→ DROP PRTRI + NEXT SORT 'H2' STO ≫ EVAL
"Area  P.  LS (triangle)"
1 10 H2 SUB
≪ { } 1 H2 SIZE FOR j H2 j GET IF DUP 1 4 SUB " 210" == THEN + END NEXT ≫ EVAL

Output:
4: 517
3:   "Area  P.  LS (triangle)"
2: { "   6  12   5 (3 4 5)"
"  12  16   6 (5 5 6)"
"  12  18   8 (5 5 8)"
"  24  32  15 (4 13 15)"
"  30  30  13 (5 12 13)"
"  36  36  17 (9 10 17)"
"  36  54  26 (3 25 26)"
"  42  42  20 (7 15 20)"
"  60  36  13 (10 13 13)"
"  60  40  17 (8 15 17)" }
1: { " 210  70  28 (17 25 28)"
" 210  70  29 (20 21 29)"
" 210  84  37 (12 35 37)"
" 210  84  39 (17 28 39)"
" 210 140  68 (7 65 68)"
" 210 300 149 (3 148 149)" }


Ruby

class Triangle
def self.valid?(a,b,c)      # class method
short, middle, long = [a, b, c].sort
short + middle > long
end

def initialize(a,b,c)
@sides = [a, b, c].sort
@perimeter = a + b + c
s = @perimeter / 2.0
@area = Math.sqrt(s * (s - a) * (s - b) * (s - c))
end

def heronian?
area == area.to_i
end

def <=>(other)
[area, perimeter, sides] <=> [other.area, other.perimeter, other.sides]
end

def to_s
"%-11s%6d%8.1f" % [sides.join('x'), perimeter, area]
end
end

max, area = 200, 210
prim_triangles = []
1.upto(max) do |a|
a.upto(max) do |b|
b.upto(max) do |c|
next if a.gcd(b).gcd(c) > 1
prim_triangles << Triangle.new(a, b, c) if Triangle.valid?(a, b, c)
end
end
end

sorted = prim_triangles.select(&:heronian?).sort

puts "Primitive heronian triangles with sides upto #{max}: #{sorted.size}"
puts "\nsides       perim.   area"
puts sorted.first(10).map(&:to_s)
puts "\nTriangles with an area of: #{area}"
sorted.each{|tr| puts tr if tr.area == area}

Output:
Primitive heronian triangles with sides upto 200: 517

sides       perim.   area
3x4x5          12     6.0
5x5x6          16    12.0
5x5x8          18    12.0
4x13x15        32    24.0
5x12x13        30    30.0
9x10x17        36    36.0
3x25x26        54    36.0
7x15x20        42    42.0
10x13x13       36    60.0
8x15x17        40    60.0

Triangles with an area of: 210
17x25x28       70   210.0
20x21x29       70   210.0
12x35x37       84   210.0
17x28x39       84   210.0
7x65x68       140   210.0
3x148x149     300   210.0


Rust

use num_integer::Integer;
use std::{f64, usize};

const MAXSIZE: usize = 200;

#[derive(Debug)]
struct HerionanTriangle {
a: usize,
b: usize,
c: usize,
area: usize,
perimeter: usize,
}

fn get_area(a: &usize, b: &usize, c: &usize) -> f64 {
let s = (a + b + c) as f64 / 2.;
(s * (s - *a as f64) * (s - *b as f64) * (s - *c as f64)).sqrt()
}

fn is_heronian(a: &usize, b: &usize, c: &usize) -> bool {
let area = get_area(a, b, c);
// Heronian if the area is an integer number
area != 0. && area.fract() == 0.
}

fn main() {
let mut heronians: Vec<HerionanTriangle> = vec![];

(1..=MAXSIZE).into_iter().for_each(|a| {
(a..=MAXSIZE).into_iter().for_each(|b| {
(b..=MAXSIZE).into_iter().for_each(|c| {
if a + b > c && a.gcd(&b).gcd(&c) == 1 && is_heronian(&a, &b, &c) {
heronians.push(HerionanTriangle {
a,
b,
c,
perimeter: a + b + c,
area: get_area(&a, &b, &c) as usize,
})
}
})
})
});

// sort by area then by perimeter, then by maximum side
heronians.sort_unstable_by(|x, y| {
x.area
.cmp(&y.area)
.then(x.perimeter.cmp(&y.perimeter))
.then((x.a.max(x.b).max(x.c)).cmp(&y.a.max(y.b).max(y.c)))
});

println!(
"Primitive Heronian triangles with sides up to 200: {}",
heronians.len()
);

println!("\nFirst ten when ordered by increasing area, then perimeter,then maximum sides:");
heronians.iter().take(10).for_each(|h| println!("{:?}", h));

println!("\nAll with area 210 subject to the previous ordering:");
heronians
.iter()
.filter(|h| h.area == 210)
.for_each(|h| println!("{:?}", h));
}

Output:
Primitive Heronian triangles with sides up to 200: 517

First ten when ordered by increasing area, then perimeter,then maximum sides:
HerionanTriangle { a: 3, b: 4, c: 5, area: 6, perimeter: 12 }
HerionanTriangle { a: 5, b: 5, c: 6, area: 12, perimeter: 16 }
HerionanTriangle { a: 5, b: 5, c: 8, area: 12, perimeter: 18 }
HerionanTriangle { a: 4, b: 13, c: 15, area: 24, perimeter: 32 }
HerionanTriangle { a: 5, b: 12, c: 13, area: 30, perimeter: 30 }
HerionanTriangle { a: 9, b: 10, c: 17, area: 36, perimeter: 36 }
HerionanTriangle { a: 3, b: 25, c: 26, area: 36, perimeter: 54 }
HerionanTriangle { a: 7, b: 15, c: 20, area: 42, perimeter: 42 }
HerionanTriangle { a: 10, b: 13, c: 13, area: 60, perimeter: 36 }
HerionanTriangle { a: 8, b: 15, c: 17, area: 60, perimeter: 40 }

All with area 210 subject to the previous ordering:
HerionanTriangle { a: 17, b: 25, c: 28, area: 210, perimeter: 70 }
HerionanTriangle { a: 20, b: 21, c: 29, area: 210, perimeter: 70 }
HerionanTriangle { a: 12, b: 35, c: 37, area: 210, perimeter: 84 }
HerionanTriangle { a: 17, b: 28, c: 39, area: 210, perimeter: 84 }
HerionanTriangle { a: 7, b: 65, c: 68, area: 210, perimeter: 140 }
HerionanTriangle { a: 3, b: 148, c: 149, area: 210, perimeter: 300 }


Scala

object Heron extends scala.collection.mutable.MutableList[Seq[Int]] with App {
private final val n = 200
for (c <- 1 to n; b <- 1 to c; a <- 1 to b if gcd(gcd(a, b), c) == 1) {
val p = a + b + c
val s = p / 2D
val area = Math.sqrt(s * (s - a) * (s - b) * (s - c))
if (isHeron(area))
appendElem(Seq(a, b, c, p, area.toInt))
}
print(s"Number of primitive Heronian triangles with sides up to $n: " + length) private final val list = sortBy(i => (i(4), i(3))) print("\n\nFirst ten when ordered by increasing area, then perimeter:" + header) list slice (0, 10) map format foreach print print("\n\nArea = 210" + header) list filter { _(4) == 210 } map format foreach print private def gcd(a: Int, b: Int) = { var leftover = 1 var (dividend, divisor) = if (a > b) (a, b) else (b, a) while (leftover != 0) { leftover = dividend % divisor if (leftover > 0) { dividend = divisor divisor = leftover } } divisor } private def isHeron(h: Double) = h % 1 == 0 && h > 0 private final val header = "\nSides Perimeter Area" private def format: Seq[Int] => String = "\n%3d x %3d x %3d %5d %10d".format }  Sidef Translation of: Ruby class Triangle(a, b, c) { has (sides, perimeter, area) method init { sides = [a, b, c].sort perimeter = [a, b, c].sum var s = (perimeter / 2) area = sqrt(s * (s - a) * (s - b) * (s - c)) } method is_valid(a,b,c) { var (short, middle, long) = [a, b, c].sort...; (short + middle) > long } method is_heronian { area == area.to_i } method <=>(other) { [area, perimeter, sides] <=> [other.area, other.perimeter, other.sides] } method to_s { "%-11s%6d%8.1f" % (sides.join('x'), perimeter, area) } } var (max, area) = (200, 210) var prim_triangles = [] for a in (1..max) { for b in (a..max) { for c in (b..max) { next if (Math.gcd(a, b, c) > 1) prim_triangles << Triangle(a, b, c) if Triangle.is_valid(a, b, c) } } } var sorted = prim_triangles.grep{.is_heronian}.sort say "Primitive heronian triangles with sides upto #{max}: #{sorted.size}" say "\nsides perim. area" say sorted.first(10).join("\n") say "\nTriangles with an area of: #{area}" sorted.each{|tr| say tr if (tr.area == area)}  Output: Primitive heronian triangles with sides upto 200: 517 sides perim. area 3x4x5 12 6.0 5x5x6 16 12.0 5x5x8 18 12.0 4x13x15 32 24.0 5x12x13 30 30.0 9x10x17 36 36.0 3x25x26 54 36.0 7x15x20 42 42.0 10x13x13 36 60.0 8x15x17 40 60.0 Triangles with an area of: 210 17x25x28 70 210.0 20x21x29 70 210.0 12x35x37 84 210.0 17x28x39 84 210.0 7x65x68 140 210.0 3x148x149 300 210.0  Smalltalk Works with Squeak 5.x perimeter := [:triangle | triangle reduce: #+]. squaredArea := [:triangle | | s | s := (perimeter value: triangle) / 2. triangle inject: s into: [:a2 :edge | s - edge * a2]]. isPrimitive := [:triangle | (triangle reduce: #gcd:) = 1]. isHeronian := [:triangle | (squaredArea value: triangle) sqrt isInteger]. heroGenerator := Generator on: [:generator | 1 to: 200 do: [:a | a to: 200 do: [:b | b to: (a+b-1 min: 200) do: [:c | | triangle | triangle := {a. b. c.}. ((isPrimitive value: triangle) and: [isHeronian value: triangle]) ifTrue: [generator nextPut: triangle]]]]]. heronians := heroGenerator contents. sorter := squaredArea ascending , perimeter ascending , #third ascending , #second ascending , #first ascending. sorted := heronians sorted: sorter. area210 := sorted select: [:triangle | (squaredArea value: triangle) = 210 squared]. header := [:title | Transcript cr; cr; nextPutAll: title; cr. #(peri area a b c) do: [:s | Transcript nextPutAll: s; tab]]. tabulate := [:t | Transcript cr. Transcript print: (perimeter value: t); tab. Transcript print: (squaredArea value: t) sqrt. t do: [:edge | Transcript tab; print: edge].]. Transcript cr; print: heronians size; nextPutAll: ' heronians triangles of side <= 200 where found'. header value: 'first 10 sorted by area, then perimeter, the largest side'. (sorted first: 10) do: tabulate. header value: 'heronians of area 210'. area210 do: tabulate. Transcript flush.  Output: 517 heronians triangles of side <= 200 where found first 10 sorted by area, then perimeter, the largest side peri area a b c 12 6 3 4 5 16 12 5 5 6 18 12 5 5 8 32 24 4 13 15 30 30 5 12 13 36 36 9 10 17 54 36 3 25 26 42 42 7 15 20 36 60 10 13 13 40 60 8 15 17 heronians of area 210 peri area a b c 70 210 17 25 28 70 210 20 21 29 84 210 12 35 37 84 210 17 28 39 140 210 7 65 68 300 210 3 148 149  SPL h,t = getem(200) #.sort(h,4,5,1,2,3) #.output("There are ",t," Heronian triangles") #.output(" a b c area perimeter") #.output("----- ----- ----- ------ ---------") > i, 1..#.min(10,t) print(h,i) < #.output(#.str("...",">34<")) > i, 1..t ? h[4,i]=210, print(h,i) < print(h,i)= #.output(#.str(h[1,i],">4>")," ",#.str(h[2,i],">4>")," ",#.str(h[3,i],">4>")," ",#.str(h[4,i],">5>")," ",#.str(h[5,i],">8>")) . getem(n)= > a, 1..n > b, #.upper((a+1)/2)..a > c, a-b+1..b x = ((a+b+c)*(a+b-c)*(a-b+c)*(b-a+c))^0.5 >> x%1 | #.gcd(a,b,c)>1 t += 1 h[1,t],h[2,t],h[3,t] = #.sort(a,b,c) h[4,t],h[5,t] = heron(a,b,c) < < < <= h,t . heron(a,b,c)= s = (a+b+c)/2 <= (s*(s-a)*(s-b)*(s-c))^0.5, s*2 . Output: There are 517 Heronian triangles a b c area perimeter ----- ----- ----- ------ --------- 3 4 5 6 12 5 5 6 12 16 5 5 8 12 18 4 13 15 24 32 5 12 13 30 30 9 10 17 36 36 3 25 26 36 54 7 15 20 42 42 10 13 13 60 36 8 15 17 60 40 ... 17 25 28 210 70 20 21 29 210 70 12 35 37 210 84 17 28 39 210 84 7 65 68 210 140 3 148 149 210 300  Swift Works with Swift 1.2 import Foundation typealias PrimitiveHeronianTriangle = (s1:Int, s2:Int, s3:Int, p:Int, a:Int) func heronianArea(side1 s1:Int, side2 s2:Int, side3 s3:Int) -> Int? { let d1 = Double(s1) let d2 = Double(s2) let d3 = Double(s3) let s = (d1 + d2 + d3) / 2.0 let a = sqrt(s * (s - d1) * (s - d2) * (s - d3)) if a % 1 != 0 || a <= 0 { return nil } else { return Int(a) } } func gcd(a:Int, b:Int) -> Int { if b != 0 { return gcd(b, a % b) } else { return abs(a) } } var triangles = [PrimitiveHeronianTriangle]() for s1 in 1...200 { for s2 in 1...s1 { for s3 in 1...s2 { if gcd(s1, gcd(s2, s3)) == 1, let a = heronianArea(side1: s1, side2: s2, side3: s3) { triangles.append((s3, s2, s1, s1 + s2 + s3, a)) } } } } sort(&triangles) {t1, t2 in if t1.a < t2.a || t1.a == t2.a && t1.p < t2.p { return true } else { return false } } println("Number of primitive Heronian triangles with sides up to 200: \(triangles.count)\n") println("First ten sorted by area, then perimeter, then maximum side:\n") println("Sides\t\t\tPerimeter\tArea") for t in triangles[0...9] { println("\(t.s1)\t\(t.s2)\t\(t.s3)\t\t\(t.p)\t\t\(t.a)") }  Output: Number of primitive Heronian triangles with sides up to 200: 517 First ten sorted by area, then perimeter, then maximum side: Sides Perimeter Area 3 4 5 12 6 5 5 6 16 12 5 5 8 18 12 4 13 15 32 24 5 12 13 30 30 9 10 17 36 36 3 25 26 54 36 7 15 20 42 42 10 13 13 36 60 8 15 17 40 60  Tcl if {[info commands let] eq ""} { #make some math look nicer: proc let {name args} { tailcall ::set$name [uplevel 1 $args] } interp alias {} = {} expr namespace import ::tcl::mathfunc::* ::tcl::mathop::* interp alias {} sum {} + # a simple adaptation of gcd from http://wiki.tcl.tk/2891 proc coprime {a args} { set gcd$a
foreach arg $args { while {$arg != 0} {
set t $arg let arg =$gcd % $arg set gcd$t
if {$gcd == 1} {return true} } } return false } } namespace eval Hero { # Integer square root: returns 0 if n is not a square. proc isqrt? {n} { let r = entier(sqrt($n))
if {$r**2 ==$n} {
return $r } else { return 0 } } # The square of a triangle's area proc squarea {a b c} { let s = ($a + $b +$c) / 2.0
let sqrA = $s * ($s - $a) * ($s - $b) * ($s - $c) return$sqrA
}

proc is_heronian {a b c} {
isqrt? [squarea $a$b $c] } proc primitive_triangles {db max} { for {set a 1} {$a <= $max} {incr a} { for {set b$a} {$b <=$max} {incr b} {
let maxc = min($a+$b,$max) for {set c$b} {$c <=$maxc} {incr c} {
set area [is_heronian $a$b $c] if {$area && [coprime $a$b $c]} { set perimiter [expr {$a + $b +$c}]
$db eval {insert into herons (area, perimiter, a, b, c) values ($area, $perimiter,$a, $b,$c)}
}
}
}
}
}
}

proc main {db} {
$db eval {create table herons (area int, perimiter int, a int, b int, c int)} set max 200 puts "Calculating Primitive Heronian triangles up to size length$max"
puts \t[time {Hero::primitive_triangles $db$max} 1]

puts "Total Primitive Heronian triangles with side lengths <= $max:"$db eval {select count(1) count from herons} {
puts "\t$count" } puts "First ten when ordered by increasing area, perimiter, max side length:"$db eval {select * from herons order by area, perimiter, c limit 10} {
puts "\t($a,$b, $c) perimiter =$perimiter;  area = $area" } puts "All of area 210:"$db eval {select * from herons where area=210 order by area, perimiter, c} {
puts "\t($a,$b, $c) perimiter =$perimiter;  area = $area" } } package require sqlite3 sqlite3 db :memory: main db  Output: Calculating Primitive Heronian triangles up to size length 200 11530549 microseconds per iteration Total Primitive Heronian triangles with side lengths <= 200: 517 First ten when ordered by increasing area, perimiter, max side length: (3, 4, 5) perimiter = 12; area = 6 (5, 5, 6) perimiter = 16; area = 12 (5, 5, 8) perimiter = 18; area = 12 (4, 13, 15) perimiter = 32; area = 24 (5, 12, 13) perimiter = 30; area = 30 (9, 10, 17) perimiter = 36; area = 36 (3, 25, 26) perimiter = 54; area = 36 (7, 15, 20) perimiter = 42; area = 42 (10, 13, 13) perimiter = 36; area = 60 (8, 15, 17) perimiter = 40; area = 60 All of area 210: (17, 25, 28) perimiter = 70; area = 210 (20, 21, 29) perimiter = 70; area = 210 (12, 35, 37) perimiter = 84; area = 210 (17, 28, 39) perimiter = 84; area = 210 (7, 65, 68) perimiter = 140; area = 210 (3, 148, 149) perimiter = 300; area = 210  VBA Translation of: Phix Function heroArea(a As Integer, b As Integer, c As Integer) As Double s = (a + b + c) / 2 On Error GoTo Err heroArea = Sqr(s * (s - a) * (s - b) * (s - c)) Exit Function Err: heroArea = -1 End Function Function hero(h As Double) As Boolean hero = (h - Int(h) = 0) And h > 0 End Function Public Sub main() Dim list() As Variant, items As Integer Dim a As Integer, b As Integer, c As Integer Dim hArea As Double Dim tries As Long For a = 1 To 200 For b = 1 To a For c = 1 To b tries = tries + 1 If gcd(gcd(a, b), c) = 1 Then hArea = heroArea(a, b, c) If hero(hArea) Then ReDim Preserve list(items) list(items) = Array(CStr(hArea), CStr(a + b + c), CStr(a), CStr(b), CStr(c)) items = items + 1 End If End If Next c Next b Next a list = sort(list) Debug.Print "Primitive Heronian triangles with sides up to 200:"; UBound(list) + 1; "(of"; tries; "tested)" Debug.Print Debug.Print "First 10 ordered by area/perimeter/sides:" Debug.Print "area perimeter sides" For i = 0 To 9 Debug.Print Format(list(i)(0), "@@@"), Format(list(i)(1), "@@@"), Debug.Print list(i)(2); "x"; list(i)(3); "x"; list(i)(4) Next i Debug.Print Debug.Print "area = 210:" Debug.Print "area perimeter sides" For i = 0 To UBound(list) If Val(list(i)(0)) = 210 Then Debug.Print Format(list(i)(0), "@@@"), Format(list(i)(1), "@@@"), Debug.Print list(i)(2); "x"; list(i)(3); "x"; list(i)(4) End If Next i End Sub  Output: Primitive Heronian triangles with sides up to 200: 517 (of 1353400 tested) First 10 ordered by area/perimeter/sides: area perimeter sides 6 12 5x4x3 12 16 6x5x5 12 18 8x5x5 24 32 15x13x4 30 30 13x12x5 36 36 17x10x9 36 54 26x25x3 42 42 20x15x7 60 36 13x13x10 60 40 17x15x8 area = 210: area perimeter sides 210 70 28x25x17 210 70 29x21x20 210 84 37x35x12 210 84 39x28x17 210 140 68x65x7 210 300 149x148x3 Wren Library: Wren-math Library: Wren-sort Library: Wren-fmt import "./math" for Int, Nums import "./sort" for Sort import "./fmt" for Fmt var isInteger = Fn.new { |n| n is Num && n.isInteger } var primHeronian = Fn.new { |a, b, c| if (!(isInteger.call(a) && isInteger.call(b) && isInteger.call(c))) return [false, 0, 0] if (Int.gcd(Int.gcd(a, b), c) != 1) return [false, 0, 0] var p = a + b + c var s = p / 2 var A = (s * (s - a) * (s - b) * (s - c)).sqrt if (A > 0 && isInteger.call(A)) return [true, A, p] return [false, 0, 0] } var ph = [] for (a in 1..200) { for (b in a..200) { for (c in b..200) { var res = primHeronian.call(a, b, c) if (res[0]) { var sides = [a, b, c] ph.add([sides, res[1], res[2], Nums.max(sides)]) } } } } System.print("There are %(ph.count) primitive Heronian trangles with sides <= 200.") var cmp = Fn.new { |e1, e2| if (e1[1] != e2[1]) return (e1[1] - e2[1]).sign if (e1[2] != e2[2]) return (e1[2] - e2[2]).sign return (e1[3] - e2[3]).sign } Sort.quick(ph, 0, ph.count-1, cmp) System.print("\nThe first 10 such triangles in sorted order are:") System.print(" Sides Area Perimeter Max Side") for (t in ph.take(10)) { var sides = Fmt.swrite("$2d x $2d x$2d", t[0][0], t[0][1], t[0][2])
Fmt.print("$-14s$2d       $2d$2d", sides, t[1], t[2], t[3])
}

System.print("\nThe triangles in the previously sorted order with an area of 210 are:")
System.print("   Sides        Area   Perimeter  Max Side")
for (t in ph.where { |e| e[1] == 210 }) {
var sides = Fmt.swrite("$2d x$3d x $3d", t[0][0], t[0][1], t[0][2]) Fmt.print("$-14s   $3d$3d       \$3d", sides, t[1], t[2], t[3])
}

Output:
There are 517 primitive Heronian trangles with sides <= 200.

The first 10 such triangles in sorted order are:
Sides       Area   Perimeter  Max Side
3 x  4 x  5      6       12        5
5 x  5 x  6     12       16        6
5 x  5 x  8     12       18        8
4 x 13 x 15     24       32       15
5 x 12 x 13     30       30       13
9 x 10 x 17     36       36       17
3 x 25 x 26     36       54       26
7 x 15 x 20     42       42       20
10 x 13 x 13     60       36       13
8 x 15 x 17     60       40       17

The triangles in the previously sorted order with an area of 210 are:
Sides        Area   Perimeter  Max Side
17 x  25 x  28   210        70        28
20 x  21 x  29   210        70        29
12 x  35 x  37   210        84        37
17 x  28 x  39   210        84        39
7 x  65 x  68   210       140        68
3 x 148 x 149   210       300       149


XPL0

include xpllib;         \for Min, GCD, StrSort, StrNCmp, and Print

func Hero(A, B, C);     \Return area squared of triangle with sides A, B, C
int  A, B, C, S;
[S:= (A+B+C)/2;
if rem(0) = 1 then return 0;    \return 0 if area is not an integer
return S*(S-A)*(S-B)*(S-C);
];

func Heronian(A, B, C); \Return area of triangle if sides and area are integers
int  A, B, C, Area2, Area;
[Area2:= Hero(A, B, C);
Area:= sqrt(Area2);
return if Area*Area = Area2 then Area else 0;
];

def MaxSide = 200;
int A, B, C, Area, Count, I, J, K;
char Array(1000, 5*5);
[Format(5, 0);
Count:= 0;
for A:= 1 to MaxSide do
for B:= A to MaxSide do
for C:= B to Min(A+B-1, MaxSide) do
if GCD(GCD(B,C), A) = 1 then
[Area:= Heronian(A, B, C);
if Area > 0 then
[OpenO(8);
RlOut(8, float(Area));
RlOut(8, float(A+B+C));
RlOut(8, float(C));
RlOut(8, float(B));
RlOut(8, float(A));
OpenI(8);
for I:= 0 to 25-1 do Array(Count,I):= ChIn(8);
Count:= Count+1;
];
];
Print("Count = %d\n", Count);
StrSort(Array, Count);
Print("    A    B    C  Perim Area\n");
for I:= 0 to 10-1 do
[for J:= 4 downto 0 do
Print("%5.5s", @Array(I, J*5+K));
Print("\n");
];
Print("\n");
for I:= 0 to Count-1 do
if StrNCmp("  210", @Array(I,0), 5) = 0 then
[for J:= 4 downto 0 do
Print("%5.5s", @Array(I, J*5+K));
Print("\n");
];
]
Output:
Count = 517
A    B    C  Perim Area
3    4    5   12    6
5    5    6   16   12
5    5    8   18   12
4   13   15   32   24
5   12   13   30   30
9   10   17   36   36
3   25   26   54   36
7   15   20   42   42
10   13   13   36   60
8   15   17   40   60

17   25   28   70  210
20   21   29   70  210
12   35   37   84  210
17   28   39   84  210
7   65   68  140  210
3  148  149  300  210


zkl

Translation of: Python
fcn hero(a,b,c){  //--> area (float)
s,a2:=(a + b + c).toFloat()/2, s*(s - a)*(s - b)*(s - c);
(a2 > 0) and a2.sqrt() or 0.0
}
fcn isHeronian(a,b,c){
A:=hero(a,b,c);
(A>0) and A.modf()[1].closeTo(0.0,1.0e-6) and A  //--> area or False
}
const MAX_SIDE=200;
heros:=Sink(List);
foreach a,b,c in ([1..MAX_SIDE],[a..MAX_SIDE],[b..MAX_SIDE]){
if(a.gcd(b).gcd(c)==1 and (h:=isHeronian(a,b,c))) heros.write(T(h,a+b+c,a,b,c));
}
// sort by increasing area, perimeter, then sides
heros=heros.close().sort(fcn([(h1,p1,_,_,c1)],[(h2,p2,_,_,c2)]){
if(h1!=h2) return(h1<h2);
if(p1!=p2) return(p1<p2);
c1<c2;
});

println("Primitive Heronian triangles with sides up to %d: ".fmt(MAX_SIDE),heros.len());

println("First ten when ordered by increasing area, then perimeter,then maximum sides:");
println("Area Perimeter Sides");
heros[0,10].pump(fcn(phabc){ "%3s %8d %3dx%dx%d".fmt(phabc.xplode()).println() });

println("\nAll with area 210 subject to the previous ordering:");
println("Area Perimeter Sides");
heros.filter(fcn([(h,_)]){ h==210 })
.pump(fcn(phabc){ "%3s %8d %3dx%dx%d".fmt(phabc.xplode()).println() });
Output:
Primitive Heronian triangles with sides up to 200: 517
First ten when ordered by increasing area, then perimeter,then maximum sides:
Area Perimeter Sides
6       12   3x4x5
12       16   5x5x6
12       18   5x5x8
24       32   4x13x15
30       30   5x12x13
36       36   9x10x17
36       54   3x25x26
42       42   7x15x20
60       36  10x13x13
60       40   8x15x17

All with area 210 subject to the previous ordering:
Area Perimeter Sides
210       70  17x25x28
210       70  20x21x29
210       84  12x35x37
210       84  17x28x39
210      140   7x65x68
210      300   3x148x149