RO-TST 2010 Problem 14

Let $P$ be a point in the plane and let $\gamma$ be a circle which does not contain $P$ . Two distinct variable lines $\ell$ and $\ell'$ through $P$ meet the circle $\gamma$ at points $X$ and $Y$ , and $X'$ and $Y'$ , respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$ , respectively. Prove that the line $MN$ passes through a fixed point.

Mihai Chis