Question 2
Let $\Omega$ and $\Gamma$ be circles with centers $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C$, $M$, $N$, $D$ lie on $MN$ in that order. Let $P$ be the circumcenter of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E \neq A$ and meets $\Gamma$ again at $F \neq A$. Let $H$ be the orthocenter of triangle $PMN$.
Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$. The orthocentre of a triangle is the point of intersection of its altitudes.
Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$. The orthocentre of a triangle is the point of intersection of its altitudes.