2016 IMO Shortlist G8

Let $A_{1}, B_{1}$ and $C_{1}$ be points on sides $B C, C A$ and $A B$ of an acute triangle $A B C$ respectively, such that $A A_{1}, B B_{1}$ and $C C_{1}$ are the internal angle bisectors of triangle $A B C$. Let $I$ be the incentre of triangle $A B C$, and $H$ be the orthocentre of triangle $A_{1} B_{1} C_{1}$. Show that $$A H+B H+C H \geq A I+B I+C I$$

令$A_{1}、B_{1}$和$C_{1}$分别为锐角三角形$A B C$的边$B C、C A$和$A B$上的点，使得$A A_{1}、B B_{1}$和$C C_{1}$是三角形$A B C$的内角平分线。 Let $I$ be the incentre of triangle $A B C$, and $H$ be the orthocentre of triangle $A_{1} B_{1} C_{1}$.显示 $$A H+B H+C H \geq A I+B I+C I$$