Problem

Source: Latvian TST for Baltic Way 2022 P11

Tags: geometry, circumcircle



Let $\triangle ABC$ be an acute triangle. Point $D$ is arbitrarily chosen on the side $BC$. Let the circumcircle of the triangle $\triangle ADB$ intersect the segment $AC$ at $M$, and the circumcircle of the triangle $\triangle ADC$ intersect the segment $AB$ at $N$. Prove that the tangents of the circumcircle of the triangle $\triangle AMN$ at $M$ and $N$ intersect at a point that lies on the line $BC$.