Question 151
Let $ABC$ be a non isosceles triangle with circumcircle $(O)$ and incircle $(I)$. Denote $(O_{1})$ as the circle that is externally tangent to $(O)$ at $A'$, and also tangent to the lines $AB$, $AC$ at $A_b$, $A_c$ respectively. Define the circles $(O_{2})$, $(O_{3})$ and the points $B'$, $C'$, $B_c$, $B_a$, $C_a$, $C_b$ similarly.
1. Denote $J$ as the radical center of $(O_{1})$, $(O_{2})$, $(O_{3})$ and suppose that $JA'$ intersects $(O_{1})$ at the second point $X$, $JB'$ intersects $(O_{2})$ at the second point $Y$, $JC'$ intersects $(O_{3})$ at the second point $Z$. Prove that the circle $(XYZ)$ is tangent to $(O_{1})$, $(O_{2})$, $(O_{3})$.
2. Prove that $AA'$, $BB'$, $CC'$ are concurrent at the point $M$ and the three points $I$, $M$, $O$ are collinear.
1. Denote $J$ as the radical center of $(O_{1})$, $(O_{2})$, $(O_{3})$ and suppose that $JA'$ intersects $(O_{1})$ at the second point $X$, $JB'$ intersects $(O_{2})$ at the second point $Y$, $JC'$ intersects $(O_{3})$ at the second point $Z$. Prove that the circle $(XYZ)$ is tangent to $(O_{1})$, $(O_{2})$, $(O_{3})$.
2. Prove that $AA'$, $BB'$, $CC'$ are concurrent at the point $M$ and the three points $I$, $M$, $O$ are collinear.