Question 153
Suppose points $I$ and $H$ are the incenter and orthocenter of an acute triangle $ABC$ respectively, and points $B_1$ and $C_1$ are the midpoints of sides $AC$ and $AB$ respectively. It is known that ray $B_1I$ intersects side $AB$ at $B_2$ ($B_2 \neq B$), and ray $C_1I$ intersects the extension of $AC$ at $C_2$: $B_2C_2$ and $BC$ intersect at $K$ and $A_1$ is the circumcenter of $\triangle BHC$. Prove that three points $A$, $I$ and $A_1$ are collinear if and only if the areas of $\triangle BKB_2$ and $\triangle CKC_2$ are equal. (posed by Shen Wenxuan)