Question 71
$O$ is the circumcenter of the acute-angled triangle $ABC$. The altitudes are $AD$, $BE$ and $CF$. The line $EF$ cuts the circumcircle at $P$ and $Q$. Show that $OA$ is perpendicular to $PQ$. If $M$ is the midpoint of $BC$, show that $AP^{2}=2 AD \cdot OM$.