Question 181
Let $ABC$ be a triangle, and let $E$ and $F$ be two points lying on the lines $(AB)$ and $(AC)$ respectively, distinct from $A$, $B$ and $C$. Let also $\Omega$ be the circumcircle of $ABC$, $O$ the centre of $\Omega$, and $\Gamma$ the circumcircle of $AEF$. Finally, let $P$ be the intersection point of $\Gamma$ and $\Omega$ other than $A$, and let $Q$ be the reflection of $P$ in the line $(EF)$.
Show that $Q$ lies on the line $(BC)$ if and only if $O$ lies on the circle $\Gamma$.
Show that $Q$ lies on the line $(BC)$ if and only if $O$ lies on the circle $\Gamma$.