Question 50
Let $ABC$ be an acute, non-isosceles triangle with $I$ as its incenter. Denote $D, E$ as the points of tangency of $(I)$ on $AB, AC$, respectively. The median segments with respect to vertex $A$ of triangles $ABE$ and $ACD$ meet $(I)$ at $P, Q$, respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel CD$ respectively.
1. Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.
2. Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.
1. Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.
2. Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.