In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic. Proposed by Alex Wang