$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.
Source: Moscow Olympiad 2018, Grade 9, P4
Tags: geometry, circumcircle
$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.