Question 38
Let $ABCD$ be a convex quadrilateral with the shortest side $AB$ strictly less than the longest side $CD$. Show that there exists a point $E$ on the segment $CD$ such that for any point $P$ (different from $E$) on the segment $CD$, the length of $O_1O_2$ is constant, where $O_1$ and $O_2$ are circumcenters of the triangles $APD$ and $BPE$, respectively.