Complex ellipse

Marden's theorem

$f(z)=(z-z_A)(z-z_B)(z-z_C)$

$f'(z)=((z-20-0i)(z+10+0i)(z-20i))'=0$

$3 z^2 - (20 + 40 i) z - (200 - 200 i) = 0$

$F_1 = \frac{10}{3} ((1 + 2 i) +\sqrt {3 - 2 i})$

$F_2 = -\frac{10}{3} ((-1 - 2 i) + \sqrt{3 - 2 i}$

$\left| \frac{10}{3} ((1 + 2 i) + \sqrt {3 - 2 i})+\frac{10}{3} ((-1 -2 i) + \sqrt {3 - 2 i}\right)|^2 = \frac{400 \sqrt {13}}{9}$

Such an ellipse is called Steiner inellipse and its semi-major and semi-minor axes are given by:

$a_s, b_s = \frac 16 \sqrt{a^2+b^2+c^2 \pm 2Z}$

where $a$ , $b$ , and $c$ are the side lengths of the triangle and $Z = \sqrt{a^2+b^2+c^2 - a^2b^2-b^2c^2 - c^2a^2}$ .

From $A (0,20)$ , B (-10,0)), and $C (20,0)$ , we have $a^2 = 900$ , $b^2 = 800$ , $c^2 = 500$ , $Z = 100\sqrt{13}$ , and $a_2, b_s = \dfrac {\sqrt{2200 \pm 200\sqrt{13}}}6$ . Then we have:

$\begin{aligned} \left| F_1F_2 \right| & = 2 \sqrt{a_s^2 - b_s^2} \\ \left| F_1F_2 \right|^2 & = 4 (a_s^2 - b_s^2) \\ & = 4 \times \frac {2200+200\sqrt{13} - 2200 + 200\sqrt{13}}{36} \\ & = \frac {400\sqrt{13}}9 \end{aligned}$

Therefore, $A+B+C = 400+13+9 = \boxed{422}$ .