RO-TST 2010 Problem 18

Let $\ell$ be a line, and let $\gamma$ and $\gamma'$ be two circles. The line $\ell$ meets $\gamma$ at points $A$ and $B$ , and $\gamma'$ at points $A'$ and $B'$ . The tangents to $\gamma$ at $A$ and $B$ meet at point $C$ , and the tangents to $\gamma'$ at $A'$ and $B'$ meet at point $C'$ . The lines $\ell$ and $CC'$ meet at point $P$ . Let $\lambda$ be a variable line through $P$ and let $X$ be one of the points where $\lambda$ meets $\gamma$ , and $X'$ be one of the points where $\lambda$ meets $\gamma'$ . Prove that the point of intersection of the lines $CX$ and $C'X'$ lies on a fixed circle.

Gazeta Matematica