Question 3
Let $ABCD$ be a parallelogram whose diagonals $AC,BD$ meet at $I$. Let $G$ lie inside triangle $IAB$ and satisfy $$\angle IAG=\angle IBG=45^\circ-\frac{\angle AIB}{4}.$$ Let $E,F$ be the perpendicular projections of $C$ onto $AG$ and of $D$ onto $BG$, respectively. The median from $E$ in triangle $BEF$ and the median from $F$ in triangle $AEF$ meet at $H$.
(a) Prove that $AF,BE,IH$ are concurrent; denote their common point by $L$.
(b) Let $K=CE\cap DF$. Let $J$ be the circumcenter of $LAB$, and let $M,N$ be the circumcenters of $EIJ,FIJ$, respectively. Prove that $EM,FN$, and the line joining the circumcenters of $GAB$ and $KCD$ are concurrent.
(a) Prove that $AF,BE,IH$ are concurrent; denote their common point by $L$.
(b) Let $K=CE\cap DF$. Let $J$ be the circumcenter of $LAB$, and let $M,N$ be the circumcenters of $EIJ,FIJ$, respectively. Prove that $EM,FN$, and the line joining the circumcenters of $GAB$ and $KCD$ are concurrent.