Let be an interior point of acute triangle . Let lie on with perpendicular to . Define on and on similarly. Prove that is the circumcenter of if and only if the perimeter of is not less than any one of the perimeters of , and .
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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 $A_1B_1C_1$ is not less than any one of the perimeters of $AB_1C_1, BC_1A_1$, and $CA_1B_1$.
Izvor: Međunarodna matematička olimpijada, shortlist 2001
Školjka 
be an interior point of acute triangle 
. Let 
lie on 
with 
perpendicular to 
on 
and 
on 
similarly. Prove that 
is not less than any one of the perimeters of 
, and 
.