VN-TST 2005 Problem 1

Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$ . $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$ , tangent to $(I),(O)$ in $D,K$ (for $\omega_a$ ), $E,M$ (for $\omega_b$ ), and $F,N$ (for $\omega_c$ ).

a) Show that $DK,EM,FN$ are concurrent in a point $P$ ;

b) Show that the orthocenter of $DEF$ lies on $OP$ .