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 $CA_1B_1$.
Školjka