Question 186
Let $ABCD$ be a parallelogram and let $AC$ intersect $BD$ at $I$. Let $G$ be the point inside triangle $IAB$ that satisfies
$$
\angle IAG = \angle IBG \neq 45^\circ - \frac{\angle AIB}{4}.
$$
Let $E, F$ be the projections of $C$ on $AG$ and of $D$ on $BG$, respectively. The median from vertex $E$ of triangle $BEF$ and the median from vertex $F$ of triangle $AEF$ intersect at $H$.
a) Prove that $AF, BE$ and $IH$ are concurrent; denote the point of concurrency by $L$.
b) Let $K$ be the intersection of $CE$ and $DF$. Let $J$ be the circumcenter of triangle $LAB$ and $M, N$ be the circumcenters of $EIJ, FIJ$, respectively. Prove that $EM, FN$ and the line passing through the circumcenters of $GAB, KCD$ are concurrent.
$$
\angle IAG = \angle IBG \neq 45^\circ - \frac{\angle AIB}{4}.
$$
Let $E, F$ be the projections of $C$ on $AG$ and of $D$ on $BG$, respectively. The median from vertex $E$ of triangle $BEF$ and the median from vertex $F$ of triangle $AEF$ intersect at $H$.
a) Prove that $AF, BE$ and $IH$ are concurrent; denote the point of concurrency by $L$.
b) Let $K$ be the intersection of $CE$ and $DF$. Let $J$ be the circumcenter of triangle $LAB$ and $M, N$ be the circumcenters of $EIJ, FIJ$, respectively. Prove that $EM, FN$ and the line passing through the circumcenters of $GAB, KCD$ are concurrent.