VN-TST 2014 Problem 3

Let $ABC$ be triangle with $A<B<C$ and inscribed in a circle $(O)$ . On the minor arc $ABC$ of $(O)$ and does not contain point $A$ , choose an arbitrary point $D$ . Suppose $CD$ meets $AB$ at $E$ and $BD$ meets $AC$ at $F$ . Let $O_1$ be the incenter of triangle $EBD$ touches with $EB,ED$ and tangent to $(O)$ . Let $O_2$ be the incenter of triangle $FCD$ , touches with $FC,FD$ and tangent to $(O)$ . a) $M$ is a tangency point of $O_1$ with $BE$ and $N$ is a tangency point of $O_2$ with $CF$ . Prove that the circle with diameter $MN$ has a fixed point. b) A line through $M$ is parallel to $CE$ meets $AC$ at $P$ , a line through $N$ is parallel to $BF$ meets $AB$ at $Q$ . Prove that the circumcircles of triangles $(AMP),(ANQ)$ are all tangent to a fixed circle.