d269909faa
* fix: clean-up Project Euler 381-400 * fix: missing image extension * fix: missing subscripts Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
56 lines
1.6 KiB
Markdown
56 lines
1.6 KiB
Markdown
---
|
|
id: 5900f4ee1000cf542c510000
|
|
title: 'Problem 385: Ellipses inside triangles'
|
|
challengeType: 5
|
|
forumTopicId: 302049
|
|
dashedName: problem-385-ellipses-inside-triangles
|
|
---
|
|
|
|
# --description--
|
|
|
|
For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$.
|
|
|
|
<img class="img-responsive center-block" alt="ellipse completely insisde of triangle" src="https://cdn.freecodecamp.org/curriculum/project-euler/ellipses-inside-triangles.png" style="background-color: white; padding: 10px;">
|
|
|
|
For a given $n$, consider triangles $T$ such that:
|
|
|
|
- the vertices of $T$ have integer coordinates with absolute value $≤ n$, and
|
|
- the foci<sup>1</sup> of the largest-area ellipse inside $T$ are $(\sqrt{13}, 0)$ and $(-\sqrt{13}, 0)$.
|
|
|
|
Let $A(n)$ be the sum of the areas of all such triangles.
|
|
|
|
For example, if $n = 8$, there are two such triangles. Their vertices are (-4,-3), (-4,3), (8,0) and (4,3), (4,-3), (-8,0), and the area of each triangle is 36. Thus $A(8) = 36 + 36 = 72$.
|
|
|
|
It can be verified that $A(10) = 252$, $A(100) = 34\\,632$ and $A(1000) = 3\\,529\\,008$.
|
|
|
|
Find $A(1\\,000\\,000\\,000)$.
|
|
|
|
<sup>1</sup>The foci (plural of focus) of an ellipse are two points $A$ and $B$ such that for every point $P$ on the boundary of the ellipse, $AP + PB$ is constant.
|
|
|
|
# --hints--
|
|
|
|
`ellipsesInsideTriangles()` should return `3776957309612154000`.
|
|
|
|
```js
|
|
assert.strictEqual(ellipsesInsideTriangles(), 3776957309612154000);
|
|
```
|
|
|
|
# --seed--
|
|
|
|
## --seed-contents--
|
|
|
|
```js
|
|
function ellipsesInsideTriangles() {
|
|
|
|
return true;
|
|
}
|
|
|
|
ellipsesInsideTriangles();
|
|
```
|
|
|
|
# --solutions--
|
|
|
|
```js
|
|
// solution required
|
|
```
|