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Another Chinese projective cookie

Source: 2018 China Southeast MO Grade 10 P6

July 31, 2018
geometry

Problem Statement

In the isosceles triangle ABCABC with AB=ACAB=AC, the center of ⊙O\odot O is the midpoint of the side BCBC, and AB,ACAB,AC are tangent to the circle at points E,FE,F respectively. Point GG is on ⊙O\odot O with ∠AGE=90∘\angle AGE = 90^{\circ}. A tangent line of ⊙O\odot O passes through GG, and meets ACAC at KK. Prove that line BKBK bisects EFEF.
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