Problem

Source: 2024 USAMO Problem 5, JMO Problem 6

Tags: geometry, USAMO, USAJMO



Point $D$ is selected inside acute $\triangle ABC$ so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^{\circ} + \angle BAC$. Point $E$ is chosen on ray $BD$ so that $AE = EC$. Let $M$ be the midpoint of $BC$. Show that line $AB$ is tangent to the circumcircle of triangle $BEM$. Proposed by Anton Trygub