---
id: 5900f5431000cf542c510056
title: 'Problem 471: Triangle inscribed in ellipse'
challengeType: 5
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 < 2b < 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) = ½, r(6,2) = 1, r(12,3) = 2.

Let $G(n) = \\sum*{a=3}^n \\sum*{b=1}^{\\lfloor \\frac {a - 1} 2 \\rfloor} r(a, b)$ You are given G(10) = 20.59722222, G(100) = 19223.60980 (rounded to 10 significant digits). Find G(1011). Give your answer 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--

`euler471()` should return 1.895093981e+31.

```js
assert.strictEqual(euler471(), 1.895093981e31);
```

# --seed--

## --seed-contents--

```js
function euler471() {

  return true;
}

euler471();
```

# --solutions--

```js
// solution required
```