Question 146
An acute-angled scalene triangle $ABC$ is given, with $AC \gt BC$. Let $O$ be its circumcentre, $H$ its orthocentre, and $F$ the foot of the altitude from $C$. Let $P$ be the point (other than $A$) on the line $AB$ such that $AF = PF$, and $M$ be the midpoint of $AC$. We denote the intersection of $PH$ and $BC$ by $X$, the intersection of $OM$ and $FX$ by $Y$, and the intersection of $OF$ and $AC$ by $Z$. Prove that the points $F, M, Y$ and $Z$ are concyclic.