Problem

Source: 2021 Czech and Slovak Olympiad III A p2

Tags: geometry, circumcircle, incenter



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)