Question 20
Let $\Omega$ and $\Gamma$ be circles centred at $M$ and $N$, respectively, such that the radius of $\Omega$ is smaller than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, such that points $C, M, N$ and $D$ lie on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E \neq A$. Line $AP$ intersects $\Gamma$ again at $F \neq A$. Let $H$ be the orthocentre of triangle $PMN$.
Prove that the line passing through H parallel to AP is tangent to the circumcircle of triangle BEF.
Prove that the line passing through H parallel to AP is tangent to the circumcircle of triangle BEF.