Question 5
Let $ABCD$ be a convex quadrilateral with $\angle B\lt \angle A\lt 90^\circ$. Let $I$ be the midpoint of $AB$ and $S=AD\cap BC$. A point $R$ moves inside triangle $SAB$ with $\angle ASR=\angle BSR$. On the lines $AR,BR$, choose $E,F$, respectively, so that $BE$ and $AF$ are parallel to $RS$. Suppose $EF$ meets the circumcircle of triangle $SAB$ at $H,K$. On segment $AB$, choose $M,N$ such that $\angle AHM=\angle BHI$ and $\angle BKN=\angle AKI$.
(a) Prove that the circumcenter $J$ of triangle $SMN$ lies on a fixed line.
(b) On $BE,AF$, choose $P,Q$, respectively, so that $CP\parallel SE$ and $DQ\parallel SF$. The lines $SE,SF$ meet circle $(O)$ again at $U,V$, respectively. Let $G=AU\cap BV$. Prove that the median from $G$ in triangle $GPQ$ always passes through a fixed point.
(a) Prove that the circumcenter $J$ of triangle $SMN$ lies on a fixed line.
(b) On $BE,AF$, choose $P,Q$, respectively, so that $CP\parallel SE$ and $DQ\parallel SF$. The lines $SE,SF$ meet circle $(O)$ again at $U,V$, respectively. Let $G=AU\cap BV$. Prove that the median from $G$ in triangle $GPQ$ always passes through a fixed point.