Question 76
The points $P$ and $Q$ lie respectively on the diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ and $\frac{AP}{AC} + \frac{BQ}{BD} = 1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$ and $CNP$ are concurrent.