Question 59
Let $ABCD$ be a cyclic quadrangle, let $\gamma$ be its circumcircle, and let $M$ be the midpoint of the arc $AB$ of $\gamma$, not containing the vertices $C$ and $D$. The line through $M$ and the point where the diagonals $AC$ and $BD$ cross one another, crosses $\gamma$ again at $N$. Let $P$ and $Q$ be points on the side $CD$ such that $\angle AQD = \angle DAP$ and $\angle BPC = \angle CBQ$. Show that the circles $NPQ$ and $\gamma$ are tangent to one another.
Flavian Georgescu
Flavian Georgescu