MathDB
Triangle problem

Source: Iranian National Olympiad (3rd Round) 2002

October 1, 2006
geometrycircumcirclepower of a pointradical axisperpendicular bisectorgeometry proposed

Problem Statement

ω\omega is circumcirlce of triangle ABCABC. We draw a line parallel to BCBC that intersects AB,ACAB,AC at E,FE,F and intersects ω\omega at U,VU,V. Assume that MM is midpoint of BCBC. Let ω′\omega' be circumcircle of UMVUMV. We know that R(ABC)=R(UMV)R(ABC)=R(UMV). MEME and ω′\omega' intersect at TT, and FTFT intersects ω′\omega' at SS. Prove that EFEF is tangent to circumcircle of MCSMCS.
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