Question 181
Let $ABCD$ be a convex quadrilateral with $\angle B \lt \angle A \lt 90^\circ$. 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 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 circumcircle of $SAB$, 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.
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 circumcircle of $SAB$, 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.