Problem

Source: Romania JBMO TST 2024 Day 1 P2

Tags: geometry, square, Equilateral Triangle



Let $M$ be the midpoint of the side $AD$ of the square $ABCD.$ Consider the equilateral triangles $DFM{}$ and $BFE{}$ such that $F$ lies in the interior of $ABCD$ and the lines $EF$ and $BC$ are concurrent. Denote by $P{}$ the midpoint of $ME.$ Prove that" The point $P$ lies on the line $AC.$ The halfline $PM$ is the bisector of the angle $APF.$ Adrian Bud