SHARYGIN 2011 Problem 20

Quadrilateral $ABCD$ is circumscribed around a circle with center $I$ . Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$ . Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$ .

Nikolai Beluhov and Aleksey Zaslavsky