Problem

Source: Sharygin 2019 finals Day 1 Grade 8 P1

Tags: geometry, Sharygin Geometry Olympiad, circumcircle



A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$.