KöMaL 2011 B4391

A height of a pentagon is defined as the distance of a vertex from the opposite side. Let $P$be a pentagon with all angles equal to $108^o$ and all heights different. Show that it is possible to number the consecutive vertices of $P$ in an appropriate direction around the pentagon, such that the corresponding heights satisfy the inequalities $h_1$>$h_3$>$h_4$>$h_5$>$h_2$. (Suggested by P. Kevei and V. Vígh, Szeged)