Question 187
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $ABC$ is tangent to the sides $BC, CA, AB$ at $M, N, P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB, AC$ at $A_b, A_c$ respectively. The circles $\Omega_B, \Omega_C$ and points $B', B_a, B_c, C', C_a, C_b$ are defined similarly.
a) Prove that $B_cC_b + C_aA_c + A_bB_a \ge NP + PM + MN$.
b) Suppose $A', B', C'$ lie on $AM, BN, CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c, B_cB_a, C_aC_b$. Prove $OH$ is parallel to $IK$.
a) Prove that $B_cC_b + C_aA_c + A_bB_a \ge NP + PM + MN$.
b) Suppose $A', B', C'$ lie on $AM, BN, CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c, B_cB_a, C_aC_b$. Prove $OH$ is parallel to $IK$.