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circumcircle tangent to given circle

Source: INAMO Shortlist 2015 G8

May 14, 2019
geometrycircumcircletangent circlescircles

Problem Statement

ABCABC is an acute triangle with AB>ACAB> AC. ΓB\Gamma_B is a circle that passes through A,BA,B and is tangent to ACAC on AA. Define similar for ΓC \Gamma_C. Let DD be the intersection ΓB\Gamma_B and ΓC\Gamma_C and MM be the midpoint of BCBC. AMAM cuts ΓC\Gamma_C at EE. Let OO be the center of the circumscibed circle of the triangle ABCABC. Prove that the circumscibed circle of the triangle ODEODE is tangent to ΓB\Gamma_B.
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