NL-NWO 1991 Problem 5

Let $H$ be the orthocenter, $O$ the circumcenter, and $R$ the circumradius of an acute-angled triangle $ABC$ . Consider the circles $k_a,k_b,k_c,k_h,k$ , all with radius $R$ , centered at $A,B,C,H,M,$ respectively. Circles $k_a$ and $k_b$ meet at $M$ and $F$ ; $k_a$ and $k_c$ meet at $M$ and $E$ ; and $k_b$ and $k_c$ meet at $M$ and $D$ . $(a)$ Prove that the points $D,E,F$ lie on the circle $k_h$ . $(b)$ Prove that the set of the points inside $k_h$ that are inside exactly one of the circles $k_a,k_b,k_c$ has the area twice the area of $\triangle ABC$ .