Question 86
A convex quadrilateral $ABCD$ has the incircle $\omega$. The diagonal $AC$ intersects $\omega$ at the points $P$ and $Q$. Let $M$ and $N$ be the midpoints of the arcs $PQ$ of $\omega$ such that the points $B$ and $M$ lie in one halfplane with respect to the line $AC$ while the points $D$ and $N$ — in another.
Prove that the lines $BM$, $DN$ and $AC$ are concurrent.
Prove that the lines $BM$, $DN$ and $AC$ are concurrent.