VN-TST 2015

sorry if this has been posted before .

given a fixed circle $(O)$ and two fixed point $B,C$ on it.point A varies on circle $(O)$ . let $I$ be the midpoint of $BC$ and $H$ be the orthocenter of $\triangle ABC$ . ray $IH$ meet $(O)$ at $K$ , $AH$ meet $BC$ at $D$ , $KD$ meet $(O)$ at $M$ .a line pass $M$ and perpendicular to $BC$ meet $AI$ at $N$ . a) prove that $N$ varies on a fixed circle. b) a circle pass $N$ and tangent to $AK$ at $A$ cut $AB,AC$ at $P,Q$ . let $J$ be the midpoint of $PQ$ .prove that $AJ$ pass through a fixed point.

Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$ . The circumcircle of triangle $APB$ cuts $AC$ at $E$ , the circumcircle of triangle $APC$ cuts $AB$ at $F$ . Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$ . Let $D$ be the symmetric point of $Q$ wrt $EF$ . Angle bisector of $\angle EDF$ cuts $AP$ at $T$ . a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$ . b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$ , and $K$ the circumcenter of the triangle $DIJ$ . Straight line $DT$ cut $(K)$ at $H$ . Prove that $HK$ passes through the incenter of the triangle $DMN$ .

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