UA-UMO 2023 Problem 113

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