MathDB

circumcenter of XYZ = circumcenter of A_1B_1C_1 iff incenter = orthocenter

Source: 2018 Saudi Arabia GMO TST I p3

August 1, 2020

geometryincenterCircumcenterorthocenter

Problem Statement

Let

I, O

be the incenter, circumcenter of triangle

ABC

and

A_1, B_1, C_1

be arbitrary points on the segments

AI, BI, CI

respectively. The perpendicular bisectors of

AA_1, BB_1, CC_1

intersect each other at

X, Y

and

Z

. Prove that the circumcenter of triangle

XYZ

coincides with

O

if and only if

I

is the orthocenter of triangle

A_1B_1C_1