RO-TST 2004 Problem 8

Let $\Gamma$ be a circle, and let $ABCD$ be a square lying inside the circle $\Gamma$ . Let $\mathcal{C}_a$ be a circle tangent interiorly to $\Gamma$ , and also tangent to the sides $AB$ and $AD$ of the square, and also lying inside the opposite angle of $\angle BAD$ . Let $A'$ be the tangency point of the two circles. Define similarly the circles $\mathcal{C}_b$ , $\mathcal{C}_c$ , $\mathcal{C}_d$ and the points $B',C',D'$ respectively.

Prove that the lines $AA'$ , $BB'$ , $CC'$ and $DD'$ are concurrent.

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