Question 67
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:
a) $P$ lies on the line $AC$;
b) the halfline $PM$ is the bisector of the angle $APF$.
a) $P$ lies on the line $AC$;
b) the halfline $PM$ is the bisector of the angle $APF$.