---
id: 5900f5431000cf542c510056
title: 'Problem 471: Triangle inscribed in ellipse'
challengeType: 1
forumTopicId: 302148
dashedName: problem-471-triangle-inscribed-in-ellipse
---

# --description--

The triangle $ΔABC$ is inscribed in an ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $0 &lt; 2b &lt; a$, $a$ and $b$ integers.

Let $r(a, b)$ be the radius of the incircle of $ΔABC$ when the incircle has center $(2b, 0)$ and $A$ has coordinates $\left(\frac{a}{2}, \frac{\sqrt{3}}{2}b\right)$.

For example, $r(3, 1) = \frac{1}{2}, r(6, 2) = 1, r(12, 3) = 2$.

<img class="img-responsive center-block" alt="triangle ΔABC inscribed in an ellipse, radius of the incircle of ΔABC r(6, 2) = 1" src="https://cdn.freecodecamp.org/curriculum/project-euler/triangle-inscribed-in-ellipse-1.png" style="background-color: white; padding: 10px;" />

<img class="img-responsive center-block" alt="triangle ΔABC inscribed in an ellipse, radius of the incircle of ΔABC r(12, 3) = 2" src="https://cdn.freecodecamp.org/curriculum/project-euler/triangle-inscribed-in-ellipse-2.png" style="background-color: white; padding: 10px;" />

Let $G(n) = \sum_{a = 3}^n \sum_{b = 1}^{\left\lfloor\frac{a - 1}{2} \right\rfloor} r(a, b)$

You are given $G(10) = 20.59722222$, $G(100) = 19223.60980$ (rounded to 10 significant digits).

Find $G({10}^{11})$. Give your answer as a string in scientific notation rounded to 10 significant digits. Use a lowercase `e` to separate mantissa and exponent.

For $G(10)$ the answer would have been `2.059722222e1`

# --hints--

`triangleInscribedInEllipse()` should return a string.

```js
assert(typeof triangleInscribedInEllipse() === 'string');
```

`triangleInscribedInEllipse()` should return the string `1.895093981e31`.

```js
assert.strictEqual(triangleInscribedInEllipse(), '1.895093981e31');
```

# --seed--

## --seed-contents--

```js
function triangleInscribedInEllipse() {

  return true;
}

triangleInscribedInEllipse();
```

# --solutions--

```js
// solution required
```
