VMO 2013 Problem 6

Let $ABC$ be a cute triangle. $(O)$ is circumcircle of $\triangle ABC$ . $D$ is on arc $BC$ not containing $A$ .Line $\triangle$ moved through $H$ ( $H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$ ,circumcircle $\triangle ACH$ again at $M,N$ respectively. a.Find $\triangle$ satisfy $S_{AMN}$ max b. $d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$ ,the line through $N$ perpendicular to $DC$ respectively. $d_{1}$ cuts $d_{2}$ at $P$ .Prove that $P$ move on a fixed circle.

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