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intersection of two lines is the circumcirce of a triangle

Source: Mexican Mathematical Olympiad 2004 OMM P5

July 31, 2018
geometrycircumcircle

Problem Statement

Let ω1\omega_1 and ω2\omega_2 be two circles such that the center OO of ω2\omega_2 lies in ω1\omega_1. Let CC and DD be the two intersection points of the circles. Let AA be a point on ω1\omega_1 and let BB be a point on ω2\omega_2 such that ACAC is tangent to ω2\omega_2 in C and BC is tangent to ω1\omega_1 in CC. The line segment ABAB meets ω2\omega_2 again in EE and also meets ω1\omega_1 again in F. The line CECE meets ω1\omega_1 again in GG and the line CFCF meets the line GDGD in HH. Prove that the intersection point of GOGO and EHEH is the center of the circumcircle of the triangle DEFDEF.
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