Problem

Source: Iran 2024 3rd Round Test 1 P3

Tags: geometry, circumcircle



Consider an acute scalene triangle $\triangle{ABC}$. The interior bisector of $A$ intersects $BC$ at $E$ and the minor arc of $\overarc {BC}$ in circumcircle of $\triangle{ABC}$ at $M$. Suppose that $D$ is a point on the minor arc of $\overarc {BC}$ such that $ED=EM$. $P$ is a point on the line segment of $AD$ such that $\angle ABP=\angle ACP \not= 0$. $O$ is the circumcenter of $\triangle{ABC}$. Prove that $OP \perp AM$.