## Triangle inscribed in ellipse

### Problem 471

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

Let `r`(`a`,`b`) be the radius of the incircle of ΔABC when the incircle has center (2`b`, 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`(10^{11}).

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.