Question 94
Let $ABC$ be a triangle such that $CA \ne CB$, and let $D, F$, and $G$ be the midpoints of the sides $AB, AC$, and $BC$, respectively. A circle $\gamma$ through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. Reflect $H$ and $I$ across $F$ and $G$, respectively, to obtain $H'$ and $I'$, respectively. The line $H'I'$ meets the lines $CD$ and $FG$ at $Q$ and $M$, respectively, and the line $CM$ meets $\gamma$ again at $P$. Prove that the triangle $CPQ$ is isosceles.
IMO 2015 Shortlist
IMO 2015 Shortlist