Question 2
Let $ABC$ be an acute triangle with incircle $I$. Let $(k_a)$ be the circle whose center lies on the altitude from $A$, which passes through $A$ and is internally tangent to the incircle $(I)$ at $A_1$. Define $B_1,C_1$ analogously.
1/ Prove that $AA_1,BB_1,CC_1$ are concurrent at a point $P$.
2/ Let $(J_a),(J_b),(J_c)$ be the images of the $A$-, $B$-, and $C$-excircles of triangle $ABC$ under reflection in the midpoints of $BC,CA,AB$, respectively.
Prove that $P$ is the radical center of these three circles.
1/ Prove that $AA_1,BB_1,CC_1$ are concurrent at a point $P$.
2/ Let $(J_a),(J_b),(J_c)$ be the images of the $A$-, $B$-, and $C$-excircles of triangle $ABC$ under reflection in the midpoints of $BC,CA,AB$, respectively.
Prove that $P$ is the radical center of these three circles.