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Two triangles have the same nine point centers

Source: 2021 Oral Moscow Geometry Olympiad grades 10-11 p5

May 17, 2021

geometryincentercircumcircle

Problem Statement

Let

ABC

be a triangle,

I

and

O

be its incenter and circumcenter respectively.

A'

is symmetric to

O

with respect to line

AI

. Points

B'

and

C'

are defined similarly. Prove that the nine-point centers of triangles

ABC

and

A'B'C'

coincide.