Question 90
Let $\Omega$ be a circumcircle centered at $O$ of $\triangle ABC$ with $\angle B \gt 90^\circ$. Let $B_1$ be the intersection point of the line $AB$ and the tangent line to the circle $\Omega$ at the point $C$. Let $O_1$ be the circumcenter of $\triangle AB_1C$. Choose a point $B_2$ on the line segment $BB_1$ ($B_2 \neq B, B_1$). The line from $B_2$ is tangent to the circle $\Omega$ at $C_1$, closer to $C$. Let $O_2$ be the circumcenter of $\triangle AB_2C_1$.
Prove that if $OO_2 \perp AO_1$ then $A, C, O_2$ are collinear.
Prove that if $OO_2 \perp AO_1$ then $A, C, O_2$ are collinear.