Problem

Source: Ukrainian Mathematical Olympiad 2023. Day 1, Problem 11.3

Tags: geometry, tangency



In the quadrilateral $ABCD$ $\angle ABC = \angle CDA = 90^\circ$. Let $P = AC \cap BD$, $Q = AB\cap CD$, $R = AD \cap BC$. Let $\ell$ be the midline of the triangle $PQR$, parallel to $QR$. Show that the circumcircle of the triangle formed by lines $AB, AD, \ell$ is tangent to the circumcircle of the triangle formed by lines $CD, CB, \ell$. Proposed by Fedir Yudin