# Circular Domain, Complex Range!

Geometry Level 5

$$\Gamma _ 1$$ is a given circle, with points $$A, B, C$$ on the circumference. Let $$\Gamma_2$$ be the circle that passes through the midpoints of sides of triangle $$ABC$$. Let $$\ell$$ be the length of the common chord of these two circles.

For the combination which maximizes the value of $$\ell$$, what would be the corresponding value of

$\cos 2A + \cos 2B + \cos 2C \ ?$