Let
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
be an interior point of acute triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. Let
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
lie on
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
with
![OA_1](/media/m/1/d/f/1df4db2f04ac25f69bd63ec20fecf6c8.png)
perpendicular to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Define
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
on
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
and
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
on
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
similarly. Prove that
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
is the circumcenter of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
if and only if the perimeter of
![A_1B_1C_1](/media/m/1/a/f/1af9d15fbb4b582c4f99670f42359e2d.png)
is not less than any one of the perimeters of
![AB_1C_1, BC_1A_1](/media/m/5/2/0/520c86e76989a20b9b17b6f6619d2f6e.png)
, and
![CA_1B_1](/media/m/c/7/5/c750d113cb3398ae09bb5bdb76834f80.png)
.
%V0
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 $CA_1B_1$.