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ST passes through midpoint of arc ABC

Source: saint petersburg 2015,grade 11,P7

March 14, 2016
geometry proposedgeometrycircumcircleangle bisector

Problem Statement

Let BLBL be angle bisector of acute triangle ABCABC.Point KK choosen on BLBL such that ∡AKC−∡ABC=90º\measuredangle AKC-\measuredangle ABC=90º.point SS lies on the extention of BLBL from LL such that ∡ASC=90º\measuredangle ASC=90º.Point TT is diametrically opposite the point KK on the circumcircle of △AKC\triangle AKC.Prove that STST passes through midpoint of arc ABCABC.(S. Berlov) :trampoline: my 100th post :trampoline:
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