Problem

Source: Romanian ROM TST 2004, problem 8

Tags: geometry, geometric transformation, homothety, geometry proposed



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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