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CI tangent to circumcircle of BMN, where I incenter of right triangle ABC

Source: 2021 Czech and Slovak Olympiad III A p2

June 6, 2021
geometrycircumcircleincenter

Problem Statement

Let II denote the center of the circle inscribed in the right triangle ABCABC with right angle at the vertex AA. Next, denote by MM and NN the midpoints of the lines ABAB and BIBI. Prove that the line CICI is tangent to the circumscribed circle of triangle BMNBMN.
(Patrik Bak, Josef Tkadlec)
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