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common point of circles lies on Euler line

Source: Sharygin Geometry Olympiad 2016 First Round P16 grades 9-11

July 25, 2018

geometrycircumcircleEuler Line

Problem Statement

Let

BB_1

and

CC_1

be altitudes of triangle

ABC

. The tangents to the circumcircle of

AB_1C_1

B_1

and

C_1

meet AB and

AC

at points

M

and

N

respectively. Prove that the common point of circles

AMN

and

AB_1C_1

distinct from

A

lies on the Euler line of

ABC