Question 142
Let $ABC$ be a non-isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_{1})$ as the circle that is internally tangent to $(O)$ at $A_{1}$ and also tangent to segments $AB$, $AC$ at $A_{b}$, $A_{c}$ respectively. Define the circles $(O_{2})$, $(O_{3})$ and the points $B_{1}$, $C_{1}$, $B_{c}$, $B_{a}$, $C_{a}$, $C_{b}$ similarly.
1. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent at the point $M$ and the three points $I$, $M$, $O$ are collinear.
2. Prove that the circle $(I)$ is inscribed in the hexagon with 6 vertices $A_{b}$, $A_{c}$, $B_{c}$, $B_{a}$, $C_{a}$, $C_{b}$.
1. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent at the point $M$ and the three points $I$, $M$, $O$ are collinear.
2. Prove that the circle $(I)$ is inscribed in the hexagon with 6 vertices $A_{b}$, $A_{c}$, $B_{c}$, $B_{a}$, $C_{a}$, $C_{b}$.