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---
title: Heronian triangles
id: 595b98f8b5a2245e243aa831
challengeType: 5
---
## Description
< section id = 'description' >
< p >< a href = "https://en.wikipedia.org/wiki/Heron's formula" title = "wp: Heron's formula" > Hero's formula</ a > for the area of a triangle given the length of its three sides < big > a,</ big > < big > b,</ big > and < big > c</ big > is given by:</ p >< p >< big > $$A = \sqrt{s(s-a)(s-b)(s-c)},$$</ big ></ p >< p > where < big > s</ big > is half the perimeter of the triangle; that is,</ p >< p >< big > $$s=\frac{a+b+c}{2}.$$</ big ></ p >
< p > < a href = "http://www.had2know.com/academics/heronian-triangles-generator-calculator.html" title = "link: http://www.had2know.com/academics/heronian-triangles-generator-calculator.html" > Heronian triangles< / a > are triangles whose sides and area are all integers.< / p >
< p > An example is the triangle with sides 3, 4, 5 whose area is 6 (and whose perimeter is 12). < / p >
< p > 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.< / p > < p > Define a Primitive Heronian triangle as a Heronian triangle where the greatest common divisor< / p >
< p > of all three sides is 1 (unity).< / p > < p > This will exclude, for example, triangle 6, 8, 10.< / p >
Task:
< p > Implement a function based on Hero's formula that returns the first < code > n< sub > th< / sub > < / code > ordered triangles in an array of arrays.< / p >
< / section >
## Instructions
< section id = 'instructions' >
< / section >
## Tests
< section id = 'tests' >
```yml
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tests:
- text: < code > heronianTriangle</ code > is a function.
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testString: assert(typeof heronianTriangle === 'function', '< code > heronianTriangle< / code > is a function.');
- text: < code > heronianTriangle()</ code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17]]</ code >
testString: assert.deepEqual(heronianTriangle(testCases[0]), res[0], '< code > heronianTriangle()< / code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17]]< / code > ');
- text: < code > heronianTriangle()</ code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15]],</ code >
testString: assert.deepEqual(heronianTriangle(testCases[1]), res[1], '< code > heronianTriangle()< / code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15]],< / code > ');
- text: < code > heronianTriangle()</ code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15], [10, 17, 21], [7, 24, 25], [8, 29, 35], [12, 17, 25], [4, 51, 53]],</ code >
testString: assert.deepEqual(heronianTriangle(testCases[2]), res[2], '< code > heronianTriangle()< / code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15], [10, 17, 21], [7, 24, 25], [8, 29, 35], [12, 17, 25], [4, 51, 53]],< / code > ');
- text: < code > heronianTriangle()</ code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15], [10, 17, 21], [7, 24, 25], [8, 29, 35], [12, 17, 25], [4, 51, 53], [19, 20, 37],[16, 17, 17], [17, 17, 30], [16, 25, 39], [13, 20, 21]]</ code >
testString: assert.deepEqual(heronianTriangle(testCases[3]), res[3], '< code > heronianTriangle()< / code > should return < code > [[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15], [10, 17, 21], [7, 24, 25], [8, 29, 35], [12, 17, 25], [4, 51, 53], [19, 20, 37],[16, 17, 17], [17, 17, 30], [16, 25, 39], [13, 20, 21]]< / code > ');
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```
< / section >
## Challenge Seed
< section id = 'challengeSeed' >
< div id = 'js-seed' >
```js
// noprotect
function heronianTriangle (n) {
// Good luck!
return [];
}
```
< / div >
### After Test
< div id = 'js-teardown' >
```js
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const testCases = [10, 15, 20, 25];
const res = [
[[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17]],
[[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15]],
[[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15], [10, 17, 21], [7, 24, 25], [8, 29, 35], [12, 17, 25], [4, 51, 53]],
[[3, 4, 5], [5, 5, 6], [5, 5, 8], [4, 13, 15], [5, 12, 13], [9, 10, 17], [3, 25, 26], [7, 15, 20], [10, 13, 13], [8, 15, 17], [13, 13, 24], [6, 25, 29], [11, 13, 20], [5, 29, 30], [13, 14, 15], [10, 17, 21], [7, 24, 25], [8, 29, 35], [12, 17, 25], [4, 51, 53], [19, 20, 37], [16, 17, 17], [17, 17, 30], [16, 25, 39], [13, 20, 21]]
];
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```
< / div >
< / section >
## Solution
< section id = 'solution' >
```js
// noprotect
function heronianTriangle (n) {
const list = [];
const result = [];
let j = 0;
for (let c = 1; c < = 200; c++) {
for (let b = 1; b < = c; b++) {
for (let a = 1; a < = b; a++) {
if (gcd(gcd(a, b), c) === 1 & & isHeron(heronArea(a, b, c))) {
list[j++] = new Array(a, b, c, heronArea(a, b, c));
}
}
}
}
sort(list);
for (let i = 0; i < n ; i + + ) {
result[i] = [list[i][0], list[i][1], list[i][2]];
}
return result;
function heronArea(a, b, c) {
const 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) {
let leftover = 1;
let dividend = a > b ? a : b;
let divisor = a > b ? b : a;
while (leftover !== 0) {
leftover = dividend % divisor;
if (leftover > 0) {
dividend = divisor;
divisor = leftover;
}
}
return divisor;
}
function sort(arg) {
let swapped = true;
let temp = [];
while (swapped) {
swapped = false;
for (let i = 1; i < arg.length ; i + + ) {
if (arg[i][4] < arg [ i - 1 ] [ 4 ] | | arg [ i ] [ 4 ] = = = arg [ i - 1 ] [ 4 ] & & arg [ i ] [ 3 ] < arg [ i - 1 ] [ 3 ] ) {
temp = arg[i];
arg[i] = arg[i - 1];
arg[i - 1] = temp;
swapped = true;
}
}
}
}
}
```
< / section >