KöMaL 2018 B4995

Let $D$, $E$ and $F$, respectively, denote the midpoints of sides $BC$, $CA$ and $AB$ of a right-angled but not isosceles triangle $ABC$, let $O$ be the centre of the circumscribed circle of the triangle, and let $M$ be the orthocentre. The tangent drawn to the circumscribed circle at $A$ intersects line $EF$ at $P$, and the tangent drawn at point $B$ intersects line $FD$ at $Q$. Show that lines $PQ$ and $OM$ are perpendicular.