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IMO ShortList 2001, geometry problem 7

Source: IMO ShortList 2001, geometry problem 7

September 30, 2004

geometrycircumcircleperimeterperpendicularIMO Shortlist

Problem Statement

Let

O

be an interior point of acute triangle

B_1

. 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

.

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