Problem

Source: Polish National Olympiad 2016 P2- Final Round

Tags: circumscribed quadrilateral, geometry



Let $ABCD$ be a quadrilateral circumscribed on the circle $\omega$ with center $I$. Assume $\angle BAD+ \angle ADC <\pi$. Let $M, \ N$ be points of tangency of $\omega $ with $AB, \ CD$ respectively. Consider a point $K \in MN$ such that $AK=AM$. Prove that $ID$ bisects the segment $KN$.