KöMaL 2014 B4615

Each angle of the triangle ABC is smaller than $120^o$. The isogonal point of the triangle is $P$. Draw lines through point $P$ parallel to the sides. The parallels intersect side $AB$ at $D$ and $E$, side $BC$ at $F$ and $G$, and side $CA$ at $H$ and $I$. Let $K$, $L$ and $M$ denote the isogonal points of triangles DEP , FGP , HIP , respectively. Show that triangle KLM is equilateral.