Question 55
Let $A_{1}$, $B_{1}$ and $C_{1}$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_{1}$, $BB_{1}$ and $CC_{1}$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_{1}B_{1}C_{1}$. Show that
$$
AH + BH + CH \geqslant AI + BI + CI.
$$
$$
AH + BH + CH \geqslant AI + BI + CI.
$$