Problem

Source: Swiss TST 2023 P4

Tags: geometry



Let $ABC$ and $AMN$ be two similar, non-overlapping triangles with the same orientation, such that $AB=AC$ and $AM=AN$. Let $O$ be the circumcentre of the triangle $MAB$. Prove that the points $O, C, N$ and $A$ lie on a circle if and only if the triangle $ABC$ is equilateral.