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concurrent wanted, incircle, arc midpoint of circumcircle, perpendicular related

Source: 2013 Saudi Arabia Pre-TST 3.4

September 13, 2020
geometrycircumcircleconcurrencyconcurrentincircle

Problem Statement

△ABC\vartriangle ABC is a triangle with AB<BC,ΓAB < BC, \Gamma its circumcircle, KK the midpoint of the minor arc CACA of the circle CC and TT a point on Γ\Gamma such that KTKT is perpendicular to BCBC. If A′,B′A',B' are the intouch points of the incircle of △ABC\vartriangle ABC with the sides BC,ACBC,AC, prove that the lines AT,BK,A′B′AT,BK,A'B' are concurrent.
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