CZ-SK-MO 2021 Problem 2

Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$ . Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$ . Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$ .

(Patrik Bak, Josef Tkadlec)