156 lines
4.9 KiB
Markdown
156 lines
4.9 KiB
Markdown
---
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id: 595b98f8b5a2245e243aa831
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title: Heronian triangles
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challengeType: 5
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forumTopicId: 302285
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dashedName: heronian-triangles
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---
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# --description--
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[Hero's formula](https://en.wikipedia.org/wiki/Heron's formula "wp: Heron's formula") for the area of a triangle given the length of its three sides `a`, `b`, and `c` is given by:
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$A = \\sqrt{s(s-a)(s-b)(s-c)},$
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where `s` is half the perimeter of the triangle; that is,
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$s=\\frac{a+b+c}{2}.$
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Heronian triangles are triangles whose sides and area are all integers.
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An example is the triangle with sides `3, 4, 5` whose area is `6` (and whose perimeter is `12`).
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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.
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Define a Primitive Heronian triangle as a Heronian triangle where the greatest common divisor
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of all three sides is `1` (unity).
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This will exclude, for example, triangle `6, 8, 10.`
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# --instructions--
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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.
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# --hints--
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`heronianTriangle` should be a function.
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```js
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assert(typeof heronianTriangle === 'function');
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```
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`heronianTriangle(10)` should return `[[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]]`
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```js
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assert.deepEqual(heronianTriangle(testCases[0]), res[0]);
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```
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`heronianTriangle(15)` should return `[[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]],`
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```js
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assert.deepEqual(heronianTriangle(testCases[1]), res[1]);
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```
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`heronianTriangle(20)` should return `[[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]],`
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```js
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assert.deepEqual(heronianTriangle(testCases[2]), res[2]);
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```
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`heronianTriangle(25)` should return `[[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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```js
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assert.deepEqual(heronianTriangle(testCases[3]), res[3]);
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```
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# --seed--
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## --after-user-code--
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```js
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const testCases = [10, 15, 20, 25];
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const res = [
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[[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]],
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[[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]],
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[[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]],
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[[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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];
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```
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## --seed-contents--
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```js
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function heronianTriangle(n) {
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return [];
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}
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```
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# --solutions--
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```js
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function heronianTriangle(n) {
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const list = [];
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const result = [];
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let j = 0;
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for (let c = 1; c <= 200; c++) {
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for (let b = 1; b <= c; b++) {
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for (let a = 1; a <= b; a++) {
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if (gcd(gcd(a, b), c) === 1 && isHeron(heronArea(a, b, c))) {
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list[j++] = new Array(a, b, c, heronArea(a, b, c));
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}
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}
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}
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}
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sort(list);
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for (let i = 0; i < n; i++) {
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result[i] = [list[i][0], list[i][1], list[i][2]];
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}
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return result;
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function heronArea(a, b, c) {
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const s = (a + b + c) / 2;
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return Math.sqrt(s * (s - a) * (s - b) * (s - c));
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}
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function isHeron(h) { return h % 1 === 0 && h > 0; }
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function gcd(a, b) {
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let leftover = 1;
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let dividend = a > b ? a : b;
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let divisor = a > b ? b : a;
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while (leftover !== 0) {
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leftover = dividend % divisor;
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if (leftover > 0) {
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dividend = divisor;
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divisor = leftover;
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}
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}
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return divisor;
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}
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function sort(arg) {
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let swapped = true;
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let temp = [];
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while (swapped) {
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swapped = false;
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for (let i = 1; i < arg.length; i++) {
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if (arg[i][4] < arg[i - 1][4] || arg[i][4] === arg[i - 1][4] && arg[i][3] < arg[i - 1][3]) {
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temp = arg[i];
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arg[i] = arg[i - 1];
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arg[i - 1] = temp;
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swapped = true;
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}
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}
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}
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}
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}
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```
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