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euler circles tangent to a fixed circle

Source: Sharygin 2006 finals 9.2

September 12, 2019
Euler CircleNine Point Circlegeometryfixedcircle

Problem Statement

Given a circle, point AA on it and point MM inside it. We consider the chords BCBC passing through MM. Prove that the circles passing through the midpoints of the sides of all the triangles ABCABC are tangent to a fixed circle.