Question 28
Let $ABC$ be an acutangle triangle. Let $P$ and $Q$ be two points on the $[BC]$ segment. We note respectively $O_1$, $O_2$, $O_3$ and $O_4$ the centers of the circumscribed circles of the triangles $ABP$, $ABQ$, $ACP$ and $ACQ$. Show that points $O_1$, $O_2$, $O_3$ and $O_4$ are cocyclic if and only if $\widehat{BAP} = \widehat{QAC}$.