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equal segments given and wanted, circumcenters, symmetric point, perpendicular,

Source: Rioplatense Olympiad 2001 level 3 P2

September 5, 2018

geometryequal segmentsCircumcentersymmetryperpendicular

Problem Statement

Let

ABC

be an acute triangle and

A_1, B_1

and

C_1

, points on the sides

BC, CA

and

AB

, respectively, such that

CB_1 = A_1B_1

and

BC_1 = A_1C_1

. Let

D

be the symmetric of

A_1

with respect to

B_1C_1, O

and

O_1

are the circumcenters of triangles

ABC

and

A_1B_1C_1

, respectively. If

A \ne D, O \ne O_1

and

AD

is perpendicular to

OO_1

, prove that

AB = AC