Question 19
Let $C$ and $D$ be two points of the semicircle of diameter $AB$ such that $B$ and $C$ are in different half-planes with respect to the straight line $AD$. Let $M$, $N$, and $P$ denote the midpoints of $AC$, $DB$, and $CD$, respectively. Let $O_A$ and $O_B$ be the circumcenters of the triangles $ACP$ and $BDP$. Show that the lines $O_A O_B$ and $MN$ are parallel.