Heronian triangles

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

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

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

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

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

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

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

Define a Primitive Heronian triangle as a Heronian triangle where the greatest common divisor of all three sides is 1. This will exclude, for example triangle 6,8,10

The task is to:

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

Show all output here.

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

C#

<lang Csharp> 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;
                   }                
           }
       }
   }

} </lang>

Output:

<lang> 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 </lang>

D

Translation of: Python

<lang d>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));

}</lang>

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

ERRE

<lang ERRE> PROGRAM HERON

DIM LISTA%[600,4]

PROCEDURE GCD(J%,K%->MCD%)

 WHILE J%<>K% DO
    IF J%>K% THEN
        J%=J%-K%
      ELSE
        K%=K%-J%
    END IF
 END WHILE
 MCD%=J%

END PROCEDURE

BEGIN

   PRINT(CHR$(12);) !CLS
   FOR C%=1 TO 200 DO
      FOR B%=1 TO C% DO
         FOR A%=1 TO B% DO
            S#=(A%+B%+C%)/2#
            AREA#=S#*(S#-A%)*(S#-B%)*(S#-C%)
            IF AREA#>0 THEN
            AREA#=SQR(AREA#)
            IF AREA#=INT(AREA#) THEN
                GCD(B%,C%->RES%)
                GCD(A%,RES%->RES%)
                   IF RES%=1 THEN
                      COUNT%=COUNT%+1
                      LISTA%[COUNT%,0]=A%    LISTA%[COUNT%,1]=B%   LISTA%[COUNT%,2]=C%
                      LISTA%[COUNT%,3]=2*S#  LISTA%[COUNT%,4]=AREA#
                   END IF
            END IF
        END IF
    END FOR
 END FOR

END FOR

PRINT("Number of triangles:";COUNT%)

! sorting array FLIPS%=TRUE WHILE FLIPS% DO

  FLIPS%=FALSE
  FOR I%=1 TO COUNT%-1 DO
     IF LISTA%[I%,4]>LISTA%[I%+1,4] THEN
       FOR K%=0 TO 4 DO
          SWAP(LISTA%[I%,K%],LISTA%[I%+1,K%])
       END FOR
       FLIPS%=TRUE
     END IF
  END FOR

END WHILE

! first ten FOR I%=1 TO 10 DO

   PRINT(#1,LISTA%[I%,0],LISTA%[I%,1],LISTA%[I%,2],LISTA%[I%,3],LISTA%[I%,4])

END FOR PRINT

! triangle with area=210 FOR I%=1 TO COUNT% DO

   IF LISTA%[I%,4]=210 THEN
      PRINT(LISTA%[I%,0],LISTA%[I%,1],LISTA%[I%,2],LISTA%[I%,3],LISTA%[I%,4])
   END IF

END FOR END PROGRAM </lang>

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

Haskell

<lang Haskell>import qualified Data.List as L import Data.Maybe import Data.Ord import Text.Printf

-- Determine if a number n is a perfect square and return its square root if so. -- This is used instead of sqrt to avoid fixed sized floating point numbers. perfectSqrt :: Integral a => a -> Maybe a perfectSqrt n

 | n == 1    = Just 1
 | n < 4     = Nothing
 | otherwise =
 let search low high =
       let guess = (low + high) `div` 2
           square = guess ^ 2
           next
             | square == n  = Just guess
             | low == guess = Nothing
             | square < n   = search guess high
             | otherwise    = search low guess
       in next
 in search 0 n

-- Determine the area of a Heronian triangle if it is one. heronTri :: Integral a => a -> a -> a -> Maybe a heronTri a b c =

 let -- Rewrite Heron's formula to factor out the term 16 under the root.
   areaSq16    = (a + b + c) * (b + c - a) * (a + c - b) * (a + b - c)
   (areaSq, r) = areaSq16 `divMod` 16
 in if r == 0
    then perfectSqrt areaSq
    else Nothing

isPrimitive :: Integral a => a -> a -> a -> a isPrimitive a b c = gcd a (gcd b c)

third (_, _, x, _, _) = x fourth (_, _, _, x, _) = x fifth (_, _, _, _, x) = x

orders :: Ord b => [(a -> b)] -> a -> a -> Ordering orders [f] a b = comparing f a b orders (f:fx) a b =

 case comparing f a b of
   EQ -> orders fx a b
   n  -> n

main :: IO () main = do

 let range = [1 .. 200]
     tris :: [(Integer, Integer, Integer, Integer, Integer)]
     tris = L.sortBy (orders [fifth, fourth, third])
            $ map (\(a, b, c, d, e) -> (a, b, c, d, fromJust e))
            $ filter (isJust . fifth)
            [(a, b, c, a + b + c, heronTri a b c)
            | a <- range, b <- range, c <- range
            , a <= b, b <= c, isPrimitive a b c == 1]
     printTri (a, b, c, d, e) = printf "%3d %3d %3d %9d %4d\n" a b c d e
 printf "Heronian triangles found: %d\n\n" $ length tris
 putStrLn "   Sides    Perimeter Area"
 mapM_ printTri $ take 10 tris
 putStrLn ""
 mapM_ printTri $ filter ((== 210) . fifth) tris</lang>

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

Supporting implementation:

<lang J>a=:0&{"1 b=:1&{"1 c=:2&{"1 s=:(a+b+c)%2: A=:2 %: s*(s-a)*(s-b)*(s-c) P=:+/"1 isprimhero=:(0&~:*(=<.@+))@A*1=a+.b+.c

tri=: (/: A,.P,.{:"1) (#~ isprimhero)~./:"1~1+200 200 200#:i.200^3</lang>

Required examples:

<lang J> #tri 517

  10{.(,._,.A,.P) tri
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

  (#~210=A) (,._,.A,.P) tri

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</lang>

Java

<lang Java> import java.util.ArrayList; public class Heron { public static void main(String[] args) { ArrayList<int[]> list = new ArrayList<int[]>(); 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) list.add(new int[]{a, b, c, a + b + c, (int)heronArea(a, b, c)}); } } } 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; } } } } } </lang>

Output:

<lang> 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 </lang>

JavaScript

<lang JavaScript> 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("

Primitive Heronian triangles with sides up to 200: " + list.length + "

First ten when ordered by increasing area, then perimeter:

");

   for(var i = 0; i < 10; i++)

document.write(""); document.write("

Sides	Perimeter	Area
" + list[i][0] + " x " + list[i][1] + " x " + list[i][2] + "	" + list[i][3] + "	" + list[i][4] + "

Area = 210

");

   for(var i = 0; i < list.length; i++)

if(list[i][4] == 210)

document.write("");

   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){

} </lang>

Output:

<lang> 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 </lang>

jq

Works with: jq version 1.4

<lang jq># 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)</lang>

Output:

<lang sh>$ 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</lang>

Nim

<lang nim>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{:3.3f}\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</lang>

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

Derived from REXX with some changes <lang rexx>/*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)</lang>

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

Perl

Translation of: Perl 6

<lang perl>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();</lang>

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

Works with: rakudo version 2015-01-03

<lang perl6>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 {

   say "   Area Perimeter   Sides";
   for @_ -> [$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;

}</lang>

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

PowerShell

<lang powershell> function Get-Gcd($a, $b){

   if($a -ge $b){
       $dividend = $a
       $divisor = $b
   }
   else{
       $dividend = $b
       $divisor = $a
   }
   $leftover = 1
   while($leftover -ne 0){
       $leftover = $dividend % $divisor
       if($leftover -ne 0){

$dividend = $divisor $divisor = $leftover }

   }
   $divisor

} function Is-Heron($heronArea){

   $heronArea -gt 0 -and $heronArea % 1 -eq 0

} function Get-HeronArea($a, $b, $c){

   $s = ($a + $b + $c) / 2
   [math]::Sqrt($s * ($s - $a) * ($s - $b) * ($s - $c))

} $result = @() foreach ($c in 1..200){

   for($b = 1; $b -le $c; $b++){
       for($a = 1; $a -le $b; $a++){
           if((Get-Gcd $c (Get-Gcd $b $a)) -eq 1 -and (Is-Heron(Get-HeronArea $a $b $c))){
               $result += @(,@($a, $b, $c,($a + $b + $c), (Get-HeronArea $a $b $c)))                
           }
       }
   }

} $result = $result | sort-object @{Expression={$_[4]}}, @{Expression={$_[3]}}, @{Expression={$_[2]}} "Primitive Heronian triangles with sides up to 200: $($result.length)`nFirst ten when ordered by increasing area, then perimeter,then maximum sides:`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])"
   }

} </lang>

Output:

<lang> 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 </lang>

Python

<lang python>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))</lang>

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

Racket

<lang>#lang racket (require xml/xml data/order)

Returns the area of triangle sides a, b, c

(define (A a b c)

 (define s (/ (+ a b c) 2)) ; where s=\frac{a+b+c}{2}.
 (sqrt (* s (- s a) (- s b) (- s c)))) ; A = \sqrt{s(s-a)(s-b)(s-c)}

Returns same as A iff a, b, c and A are integers; #f otherwise

(define (heronian?-area a b c)

 (and (integer? a) (integer? b) (integer? c)
      (let ((h (A a b c))) (and (integer? h) h))))

Returns same as heronian?-area, with the additional condition that (gcd a b c) = 1

(define (primitive-heronian?-area a b c)

 (and (= 1 (gcd a b c))
      (heronian?-area a b c)))

(define (generate-heronian-triangles max-side)

 (for*/list
     ((a (in-range 1 (add1 max-side)))
      (b (in-range 1 (add1 a)))
      (c (in-range 1 (add1 b)))
      #:when (< a (+ b c))
      (h (in-value (primitive-heronian?-area a b c)))
      #:when h)
   (define rv (vector h (+ a b c) (sort (list a b c) >))) ; datum-order can sort this for the tables
   rv))

Order the triangles by first increasing area, then by increasing perimeter, then by increasing maximum side lengths

(define (tri<? t1 t2)

 (eq? '< (datum-order t1 t2)))

(define triangle->tds (match-lambda [`#(,h ,p ,s) `((td ,(~a s)) (td ,(~a p)) (td ,(~a h)))]))

(define (triangles->table ts)

 `(table
   (tr (th "#") (th "sides") (th "perimiter") (th "area")) "\n"
   ,@(for/list ((i (in-naturals 1)) (t ts)) `(tr (td ,(~a i)) ,@(triangle->tds t) "\n"))))

(define (sorted-triangles-table triangles)

 (triangles->table (sort triangles tri<?)))

(module+ main

 (define ts (generate-heronian-triangles 200))
 (define div-out
   `(div
     (p "number of primitive triangles found with perimeter "le" 200 = " ,(~a (length ts))) "\n"
     ;; Show the first ten ordered triangles in a table of sides, perimeter, and area.
     ,(sorted-triangles-table (take (sort ts tri<?) 10)) "\n"
     ;; Show a similar ordered table for those triangles with area = 210
     ,(sorted-triangles-table (sort (filter (match-lambda [(vector 210 _ _) #t] [_ #f]) ts) tri<?))))
 
 (displayln (xexpr->string div-out)))</lang>

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

Sides	Perimeter	Area
" + list[i][0] + " x " + list[i][1] + " x " + list[i][2] + "	" + list[i][3] + "	" + list[i][4] + "

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

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

REXX

Programming notes:

The hGCD subroutine is a specialized version of a GCD routine in that it doesn't check for non-positive integers.

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.

Note that the iSQRT subroutine doesn't use floating point.

The REXX statement (line 22) if pos(.,_)\==0 isn't normally used for a general-purpose check if a number is an integer (or not),
this version was used because it's faster than the usual statement: if \datatype(_,'Whole') and because the number is well-formed.

This REXX program doesn't need to explicitly sort the triangles. <lang rexx>/*REXX pgm generates primitive Heronian triangles by side length & area.*/ parse arg N first area . /*get optional N (sides). */ if N== | N==',' then N=200 /*maybe 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*/ call Heron /*invoke Heron subroutine. */ say # ' primitive Heronian triangles found with sides up to ' N " (inclusive)." call show , 'listing of the first ' first ' primitive Heronian triangles:' call show area, 'listing of the (above) found primitive Heronian triangles with an area of ' area exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────HERON subroutine────────────────────*/ Heron: @.=.; #=0; minP=9e9; maxP=0; minA=9e9; maxA=0; Ln=length(N)

                        #.=0;  #.2=1  #.3=1;  #.7=1;  #.8=1  /*¬good √.*/
 do     a=3  to N                     /*start at a minimum side length.*/
 ev= \ (a//2);           inc=1+ev     /*if A is even, B & C must be odd*/
   do   b=a+ev  to N  by inc; ab=a+b  /*AB:  is used for summing below.*/
     do c=b     to N  by inc; p=ab+c;     s=p/2     /*calc Perimeter, S*/
     _=s*(s-a)*(s-b)*(s-c); if _<=0  then iterate   /*_ isn't positive.*/
     if pos(.,_)\==0                 then iterate   /*not an integer.  */
     parse var _  -1 q  ; if #.q   then iterate   /*not good square. */
     ar=iSQRT(_);     if ar*ar\==_   then iterate   /*area not integer.*/
     if hGCD(a,b,c)\==1              then iterate   /*GCD of sides ¬1. */
     #=#+1                                          /*got prim. H. tri.*/
     minP=min( p,minP);   maxP=max( p,maxP);     Lp=length(maxP)
     minA=min(ar,minA);   maxA=max(ar,maxA);     La=length(maxA);   @.ar=
     if @.ar.p.0==.  then @.ar.p.0=0;  _=@.ar.p.0+1 /*bump triangle ctr*/
     @.ar.p.0=_;  @.ar.p._=right(a,Ln) right(b,Ln) right(c,Ln) /*unique*/
     end   /*c*/                      /* [↑]  keep each unique P items.*/
   end     /*b*/
 end       /*a*/

return # /*return # of Heronian triangles.*/ /*──────────────────────────────────HGCD subroutine─────────────────────*/ hGCD: procedure; parse arg x; do j=2 for 2 /*sub handles exactly 3 args*/

     y=arg(j);  do  until _==0; _=x//y; x=y; y=_; end;   end;    return x

/*──────────────────────────────────ISQRT subroutine────────────────────*/ iSQRT: procedure; parse arg x; x=x%1; if x==0 | x==1 then return x; q=1

 do while q<=x; q=q*4; end;     r=0        /*Q will be > X at loop end.*/
 do while q>1 ; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q;end; end

return r /* R is a postive integer. */ /*──────────────────────────────────SHOW subroutine─────────────────────*/ show: m=0; say; say; parse arg ae; say arg(2); if ae\== then first=9e9 say; y=left(,9) /* [↓] skip the nothings. */

     do   i=minA  to maxA;   if @.i==.  then iterate
     if ae\== & i\==ae   then iterate    /*Area specified? Then check*/
       do j=minP  to maxP  until m>=first  /*only list  FIRST  entries.*/
       if @.i.j.0==.       then iterate    /*Not defined? Then skip it.*/
         do k=1  for @.i.j.0;   m=m+1      /*visit each perimeter entry*/
         say right(m,9) y'area:' right(i,La) y"perimeter:" right(j,Lp) y'sides:' @.i.j.k
         end   /*k*/
       end     /*j*/                       /* [↑]  use known perimeters*/
     end       /*i*/                       /* [↑]  show found triangles*/

return</lang>

Output:

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

Ruby

<lang ruby>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}</lang>

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

zkl

Translation of: Python

<lang zkl>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

}</lang> <lang zkl>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() });</lang>

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