USAMO 2024 Problem 5

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