Question 65
Let $ABC$ be a triangle such that $AB = AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$. The lines $BI$ and $CI$ meet $\Gamma$ at two new points, denoted $M$ and $N$ respectively. Let $D$ be another point of $\Gamma$, lying on the arc $BC$ not containing $A$, and let $E, F$ be, respectively, the intersections of $AD$ with $BI$ and with $CI$. Finally, let $P$ and $Q$ be, respectively, the intersections of $DM$ with $CI$ and of $DN$ with $BI$.
i. Prove that the points $D, I, P, Q$ lie on the same circle $\Omega$.
ii. Prove that the lines $CE$ and $BF$ meet on $\Omega$.
i. Prove that the points $D, I, P, Q$ lie on the same circle $\Omega$.
ii. Prove that the lines $CE$ and $BF$ meet on $\Omega$.