Question 7
Let $ABC$ be an acute scalene triangle, with orthocenter $H$ and circumcenter $O$. Its incircle $(I)$ touches $BC,CA,AB$ at $M,N,P$, respectively. Let $\Omega_A$ be a circle through $A$, externally tangent to $(I)$ at $A'$, and meeting $AB,AC$ again at $A_b,A_c$, respectively. Define $\Omega_B,\Omega_C$ and $B',B_a,B_c,C',C_a,C_b$ analogously.
a) Prove
$$B_cC_b+C_aA_c+A_bB_a\ge NP+PM+MN.$$
b) Suppose $A',B',C'$ lie on $AM,BN,CP$, respectively. Let $K$ be the circumcenter of the triangle whose side lines are $A_bA_c,B_cB_a,C_aC_b$. Prove that $OH\parallel IK$.
a) Prove
$$B_cC_b+C_aA_c+A_bB_a\ge NP+PM+MN.$$
b) Suppose $A',B',C'$ lie on $AM,BN,CP$, respectively. Let $K$ be the circumcenter of the triangle whose side lines are $A_bA_c,B_cB_a,C_aC_b$. Prove that $OH\parallel IK$.