Question 175
Let $ABCD$ be a convex quadrilateral. Ray $AD$ meets ray $BC$ at $P$. Let $O, O'$ be the circumcenters of triangles $PCD, PAB$, respectively, and $H, H'$ be the orthocenters of triangles $PCD, PAB$, respectively. Prove that the circumcircle of triangle $DOC$ is tangent to the circumcircle of triangle $AO'B$ if and only if the circumcircle of triangle $DHC$ is tangent to the circumcircle of triangle $AH'B$.