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Tangent geo with ratios

Source: Iran RMM TST 2019, day1 p1

July 30, 2019
geometryratio

Problem Statement

Let ABCABC be a triangle and DD be the feet of AA -altitude.\\ E,FE,F are defined on segments AD,BCAD,BC ,respectively such that AEDE=BFCF\frac {AE}{DE}=\frac{BF}{CF} .\\ Assume that GG lies on AFAF such that BG⊥AFBG\perp AF .Prove that EFEF is tangent to the circumcircle of CFGCFG .
Proposed by Mehdi Etesami Fard
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