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Geometry from Iranian TST 2017

Source: Iranian TST 2017, first exam, day1, problem 3

April 5, 2017
geometryIranIranian TSTcollinear

Problem Statement

In triangle ABCABC let IaI_a be the AA-excenter. Let ω\omega be an arbitrary circle that passes through A,IaA,I_a and intersects the extensions of sides AB,ACAB,AC (extended from B,CB,C) at X,YX,Y respectively. Let S,TS,T be points on segments IaB,IaCI_aB,I_aC respectively such that ∠AXIa=∠BTIa\angle AXI_a=\angle BTI_a and ∠AYIa=∠CSIa\angle AYI_a=\angle CSI_a.Lines BT,CSBT,CS intersect at KK. Lines KIa,TSKI_a,TS intersect at ZZ. Prove that X,Y,ZX,Y,Z are collinear.
Proposed by Hooman Fattahi
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