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UK has a fixed point when isogonal points P ,Q move , many projections

Source: 2015 VMEO IV Seniors 11.2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

September 19, 2021
geometryprojectionsFixed pointfixedisogonal

Problem Statement

Let ABCABC be a triangle with two isogonal points P P and QQ . Let D,ED, E be the projection of PP on ABAB, ACAC. GG is the projection of QQ on BCBC. UU is the projection of GG on DEDE, L L is the projection of PP on AQAQ, KK is the symmetric of LL wrt UGUG. Prove that UKUK passes through a fixed point when PP and QQ vary.
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