VN-TST 2023 Problem 5

Let $ABCD$ be a convex quadrilateral with $\angle B < \angle A < 90^{o}$ . Let $I$ be the midpoint of $AB$ and $S$ the intersection of $AD$ and $BC$ . Let $R$ be a variable point inside the triangle $SAB$ such that $\angle ASR = \angle BSR$ . On the straight lines $AR, BR$ , take the points $E, F$ , respectively so that $BE , AF$ are parallel to $RS$ . Suppose that $EF$ intersects the circumcircle of triangle $SAB$ at points $H, K$ . On the segment $AB$ , take points $M , N$ such that $\angle AHM =\angle BHI$ , $\angle BKN = \angle AKI$ .

a) Prove that the center $J$ of the circumcircle of triangle $SMN$ lies on a fixed line.

b) On $BE, AF$ , take the points $P, Q$ respectively so that $CP$ is parallel to $SE$ and $DQ$ is parallel to $SF$ . The lines $SE, SF$ intersect the circle $(SAB)$ , respectively, at $U, V$ . Let $G$ be the intersection of $AU$ and $BV$ . Prove that the median of vertex $G$ of the triangle $GPQ$ always passes through a fixed point .