Question 27
Let $ABC$ be a triangle and $O$ the center of its circumcircle. Let $d$ be the line parallel to $(BC)$ through $O$. Let $A'$ be the reflection of $A$ across $(BC)$. The line parallel to $(A'B)$ through $C$ meets $d$ at $C_1$, and the lines $(A'C)$ and $(BC_1)$ intersect at $C_2$. The line parallel to $(A'C)$ through $B$ meets $d$ at $B_1$, and the lines $(A'B)$ and $(CB_1)$ intersect at $B_2$. Show that the points $A$, $A'$, $B_2$, $C_2$ are concyclic.