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line touches a fixed circle., perpendicular bisectors and circumcircle related

Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p6

August 16, 2020

geometryperpendicular bisectorcircumcirclefixedcircle

Problem Statement

Perpendicular bisectors of the sides

BC

and

AC

of an acute-angled triangle

ABC

intersect lines

AC

and

BC

at points

M

and

N

. Let point

C

move along the circumscribed circle of triangle

ABC

, remaining in the same half-plane relative to

AB

(while points

A

and

B

are fixed). Prove that line

MN

touches a fixed circle.