Heronian triangles

From Rosetta Code
Task
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:

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


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.


Task
  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


ALGOL 68[edit]

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[edit]

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[edit]

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)
script heading
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

AutoHotkey[edit]

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"
. "`n`nFirst 10 results:"
. "`n" "Area`tPerimeter`tSides`n" res2
. "`nResults for Area = 210:"
. "`n" "Area`tPerimeter`tSides`n" 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[edit]

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.

 
/*Abhishek Ghosh, 6th October 2017*/
 
#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;
 
void addAndOrderList(list *a,triangle t){
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){
addAndOrderList(&herons,t);
(*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++[edit]

Works with: C++11
#include <algorithm>
#include <cmath>
#include <iostream>
#include <tuple>
#include <vector>
 
int gcd(int a, int b)
{
int rem = 1, dividend, divisor;
std::tie(divisor, dividend) = std::minmax(a, b);
while (rem != 0) {
rem = dividend % divisor;
if (rem != 0) {
dividend = divisor;
divisor = rem;
}
}
return divisor;
}
 
struct Triangle
{
int a;
int b;
int c;
};
 
int perimeter(const Triangle& triangle)
{
return triangle.a + triangle.b + triangle.c;
}
 
double area(const Triangle& t)
{
double p_2 = perimeter(t) / 2.;
double area_sq = p_2 * ( p_2 - t.a ) * ( p_2 - t.b ) * ( p_2 - t.c );
return sqrt(area_sq);
}
 
std::vector<Triangle> 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};
double t_area = area(t);
if(t_area == 0) continue;
if( std::floor(t_area) == std::ceil(t_area) && gcd(a, gcd(b, c)) == 1)
result.push_back(t);
}
return result;
}
 
bool compare(const Triangle& lhs, const Triangle& rhs)
{
return std::make_tuple(area(lhs), perimeter(lhs), std::max(lhs.a, std::max(lhs.b, lhs.c))) <
std::make_tuple(area(rhs), perimeter(rhs), std::max(rhs.a, std::max(rhs.b, rhs.c)));
}
 
struct area_compare
{
bool operator()(const Triangle& t, int i) { return area(t) < i; }
bool operator()(int i, const Triangle& t) { return i < area(t); }
};
 
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 << area(tri[i]) << '\t' << perimeter(tri[i]) << "\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 << area(*it) << '\t' << perimeter(*it) << "\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

C#[edit]

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

CoffeeScript[edit]

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[edit]

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

EchoLisp[edit]

 
;; 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[edit]

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[edit]

 
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

Fortran[edit]

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[edit]

' 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[edit]

 
include "ConsoleWindow"
 
// Set width of tabs
def tab 10
 
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
 

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[edit]

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[edit]

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[edit]

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&~: * (= <[email protected]:+))@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&~: *. (= <[email protected]:+)
isPrimHero=: [email protected] *. 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[edit]

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[edit]

Imperative[edit]

 
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)[edit]

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[edit]

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 task(maxside):
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[edit]

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[edit]

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

Lua[edit]

-- 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

Nim[edit]

import math, algorithm, strfmt, sequtils
 
type
HeronianTriangle = tuple
a: int
b: int
c: int
s: float
A: int
 
proc `$` (t: HeronianTriangle): string =
fmt("{:3d}, {:3d}, {:3d}\t{:7.3f}\t{:3d}", t.a, t.b, t.c, t.s, t.A)
 
proc hero(a:int, b:int, c:int): tuple[s, A: float] =
let s: float = (a + b + c) / 2
result = (s, sqrt( s * (s - float(a)) * (s - float(b)) * (s - float(c)) ))
 
proc isHeronianTriangle(x: float): bool = ceil(x) == x and x.toInt > 0
 
proc gcd(x: int, y: int): int =
var
(dividend, divisor) = if x > y: (x, y) else: (y, x)
remainder = dividend mod divisor
 
while remainder != 0:
dividend = divisor
divisor = remainder
remainder = dividend mod divisor
result = divisor
 
 
var list = newSeq[HeronianTriangle]()
const max = 200
 
for c in 1..max:
for b in 1..c:
for a in 1..b:
let (s, A) = hero(a, b, c)
if isHeronianTriangle(A) and gcd(a, gcd(b, c)) == 1:
let t:HeronianTriangle = (a, b, c, s, A.toInt)
list.add(t)
 
echo "Numbers of Heronian Triangle : ", list.len
 
list.sort do (x, y: HeronianTriangle) -> int:
result = cmp(x.A, y.A)
if result == 0:
result = cmp(x.s, y.s)
if result == 0:
result = cmp(max(x.a, x.b, x.c), max(y.a, y.b, y.c))
 
echo "Ten first Heronian triangle ordered : "
echo "Sides Perimeter Area"
for t in list[0 .. <10]:
echo t
 
echo "Heronian triangle ordered with Area 210 : "
echo "Sides Perimeter Area"
for t in list.filter(proc (x: HeronianTriangle): bool = x.A == 210):
echo t
Output:
Numbers of Heronian Triangle : 517
Ten first Heronian triangle ordered : 
Sides          Perimeter Area
  3,   4,   5	  6.000	  6
  5,   5,   6	  8.000	 12
  5,   5,   8	  9.000	 12
  4,  13,  15	 16.000	 24
  5,  12,  13	 15.000	 30
  9,  10,  17	 18.000	 36
  3,  25,  26	 27.000	 36
  7,  15,  20	 21.000	 42
 10,  13,  13	 18.000	 60
  8,  15,  17	 20.000	 60
Heronian triangle ordered with Area 210 : 
Sides          Perimeter Area
 17,  25,  28	 35.000	210
 20,  21,  29	 35.000	210
 12,  35,  37	 42.000	210
 17,  28,  39	 42.000	210
  7,  65,  68	 70.000	210
  3, 148, 149	150.000	210

ooRexx[edit]

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[edit]

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[edit]

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[edit]

Translation of: Perl 6
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

Perl 6[edit]

Works with: rakudo version 2015-10-26
sub hero($a, $b, $c) {
my $s = ($a + $b + $c) / 2;
my $a2 = $s * ($s - $a) * ($s - $b) * ($s - $c);
$a2.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 @h = sort gather
for 1 .. $maxside -> $c {
for 1 .. $c -> $b {
for $c - $b + 1 .. $b -> $a {
if primitive-heronian-area($a,$b,$c) -> $area {
take [$area, $a+$b+$c, $c, $b, $a];
}
}
}
}
 
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

Phix[edit]

function heroArea(integer a, b, c)
atom s = (a+b+c)/2
return sqrt(s*(s-a)*(s-b)*(s-c))
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")
for i=1 to 10 do
printf(1,"\n%4d  %3d  %dx%dx%d",list[i])
end for
printf(1,"\n\narea = 210:\n")
printf(1,"area perimeter sides")
for i=1 to length(list) do
if list[i][1]=210 then
printf(1,"\n%4d  %3d  %dx%dx%d",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[edit]

 
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:`nSides`t`t`t`tPerimeter`tArea"
for($i = 0; $i -lt 10; $i++){
"$($result[$i][0])`t$($result[$i][1])`t$($result[$i][2])`t`t`t$($result[$i][3])`t`t`t$($result[$i][4])"
}
"`nArea = 210`nSides`t`t`t`tPerimeter`tArea"
foreach($i in $result){
if($i[4] -eq 210){
"$($i[0])`t$($i[1])`t$($i[2])`t`t`t$($i[3])`t`t`t$($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[edit]

from __future__ import division, print_function
from math import sqrt
from fractions import gcd
from itertools import product
 
 
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
h = [(a, b, c) for a,b,c in product(range(1, maxside + 1), repeat=3)
if a <= b <= c and a + b > c and gcd3(a, b, c) == 1 and is_heronian(a, b, c)]
h.sort(key = lambda x: (hero(*x), sum(x), x[::-1])) # By increasing area, perimeter, then sides
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[edit]

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[edit]

#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



REXX[edit]

using iSQRT[edit]

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; 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; 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. */
even= a//2==0 /*if A is even, B and C must be odd.*/
do b=a+even to N by 1+even; ab=a + b /*AB: is a "shortcut" sum. */
if b//2==0 then bump=1 /*B is even? Then C is odd. */
else if even then bump=0 /*A is even? " " " " */
else bump=1 /*A & B odd, then biz as usual. */
do c=b+bump to N by 2; s=(ab+c)%2 /*calculate ½ of the 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)
[email protected].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 number of Heronian triangles. */
/*────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
hGCD: procedure; parse arg x; do j=2 for 2; y=arg(j); do until y==0; parse value x//y y with y x; end; end; return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
hIsqrt: procedure; parse arg x; q=1; r=0; do while q<=x; q=q*4; end; do while q>1
q=q%4; _=x-r-q; r=r%2; if _>=0 then parse value _ r+q with x r; end; 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*/ /* [↑] show any found triangles. */
return
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[edit]

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 over seven 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=5 to N**2%2; _=i*i; !._=i; end /*pre-calculate a fast √ */
do a=3 to N /*start at a minimum side length of 3. */
even= (a//2==0) /*if A is even, B and C must be odd.*/
do b=a+even to N by 1+even; ab=a + b /*AB: is a shortcut sum. */
if b//2==0 then bump=1 /*B is even? Then C is odd. */
else if even then bump=0 /*A is even? " " " " */
else bump=1 /*A & B odd, biz as usual. */
do c=b+bump to N by 2; s=(ab + c) % 2 /*calculate perimeter: S. */
_=s*(s-a)*(s-b)*(s-c); if !._==. then iterate /*Is _ not a square? Skip.*/
ar=!._; if ar*ar\==_ then iterate /*Area not an integer? Skip.*/
if hGCD(a,b,c) \== 1 then iterate /*GCD of sides not 1? Skip.*/
#=# + 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=
[email protected].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: procedure; parse arg x; do j=2 for 2; y=arg(j); do until y==0; parse value x//y y with y x; end; end; 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*/ /* [↑] show any found triangles. */
return
output   is identical to the 1st REXX version.


Ruby[edit]

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

Scala[edit]

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[edit]

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

SPL[edit]

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[edit]

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[edit]

 
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

zkl[edit]

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