Problem

Source: Iran MO 2023 3rd round , geometry exam P2

Tags: geometry



In triangle $\triangle ABC$ , $M$ is the midpoint of arc $(BAC)$ and $N$ is the antipode of $A$ in $(ABC)$. The line through $B$ perpendicular to $AM$ , intersects $AM , (ABC)$ at $D,P$ respectively and a line through $D$ perpendicular to $AC$ , intersects $BC,AC$ at $F,E$ respectively. Prove that $PE,MF,ND$ are concurrent.