Question 21
Let O be an interior point of acute triangle ABC. Let $A_{1}$ lie on $BC$ with $OA_{1}$ perpendicular to $BC$. Define $B_{1}$ on CA and $C_{1}$ on $AB$ similarly. Prove that O is the circumcenter of $ABC$ if and only if the perimeter of $A_{1}B_{1}C_{1}$ is not less than any one of the perimeters of $AB_{1}C_{1}$, $BC_{1}A_{1}$, and $CA_{1}B_{1}$.