VN-TST 2016 Problem 3

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$ , $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$ . The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ( $G \ne A$ ). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$ . Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$ . On $EF$ take $Q$ for $QP = QI$ . Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$ . Prove that the midpoint of $IJ$ belongs to a fixed circle.

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