Question 136
Let $ABC$ be an acute triangle, and let $A_1, B_1, C_1$ lie on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incenter of triangle $ABC$ and $H$ the orthocenter of triangle $A_1B_1C_1$. Prove that
$$
AH + BH + CH \ge AI + BI + CI.
$$
$$
AH + BH + CH \ge AI + BI + CI.
$$