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right angle wanted, circumcircle and another circle related, angle bisector

Source: 2018 Saudi Arabia IMO TST IV p2

July 27, 2020
geometrycircumcircleright angleangle bisector

Problem Statement

Let ABCABC be an acute-angled triangle inscribed in circle (O)(O). Let GG be a point on the small arc ACAC of (O)(O) and (K)(K) be a circle passing through AA and GG. Bisector of ∠BAC\angle BAC cuts (K)(K) again at PP. The point EE is chosen on (K)(K) such that AEAE is parallel to BCBC. The line PKPK meets the perpendicular bisector of BCBC at FF. Prove that ∠EGF=90o\angle EGF = 90^o.
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