Let $ABC$ be an acute, non isosceles triangle and $(O)$ be its circumcircle (with center $O$). Denote by $G$ the centroid of the triangle $ABC$, by $H$ the foot of the altitude from $A$ onto the side $BC$ and by $I$ the midpoint of $AH$. The line $IG$ intersects $BC$ at $K$. 1. Prove that $CK = BH$. 2. The ray $(GH$ intersects $(O)$ at L. Denote by $T$ the circumcenter of the triangle $BHL$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.
Problem
Source: 2017 Saudi Arabia JBMO TST 1.4
Tags: geometry, circumcircle, concurrent, equal segments