Question 23
Let $ABCD$ be a convex quadrilateral with pairwise distinct side lengths such that $AC \perp BD$. Let $O_1, O_2$ be the circumcenters of $\triangle ABD, \triangle CBD$, respectively. Show that $AO_2, CO_1$, the Euler line of $\triangle ABC$ and the Euler line of $\triangle ADC$ are concurrent. (Remark. The Euler line of a triangle is the line on which its circumcenter, centroid, and orthocenter lie.)