A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.
Problem
Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade
Tags: geometry, Inscribed circle, quadrilateral