Question 165
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 center of the circle $(BHL)$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.
1. Prove that $CK = BH$.
2. The ray $GH$ intersects $(O)$ at $L$. Denote by $T$ the center of the circle $(BHL)$. Prove that $AO$ and $BT$ intersect on the circle $(O)$.