2019 OIM Problems/Problem 3

Problem

Let $\Gamma$ be the circumcircle of triangle $ABC$ . The parallel to $AC$ that passes through $B$ cuts $\Gamma$ at $D$ ( $D \ne B$ ) and the parallel to $AB$ that passes through $C$ cuts $\Gamma$ at $E$ ( $E \ne C$ ). The straight lines $AB$ and $CD$ intersect at $P$ , and lines $AC$ and $BE$ intersect at $Q$ . Let $M$ be the midpoint of $DE$ . The line $AM$ cuts $\Gamma$ at $Y$ ( $Y \ne A$ ) and the line $PQ$ at $J$ . The line $PQ$ cuts the circumcircle of triangle $BCJ$ at $Z$ ( $Z \ne J$ ). If the lines $BQ$ and $CP$ intersect at $X$ , show that $X$ belongs to the line $YZ$ .