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Right angles on incircle

Source: RMO 2018 P6

October 7, 2018
geometryincenter

Problem Statement

Let ABCABC be an acute-angled triangle with AB<ACAB<AC. Let II be the incentre of triangle ABCABC, and let D,E,FD,E,F be the points where the incircle touches the sides BC,CA,AB,BC,CA,AB, respectively. Let BI,CIBI,CI meet the line EFEF at Y,XY,X respectively. Further assume that both XX and YY are outside the triangle ABCABC. Prove that (i)\text{(i)} B,C,Y,XB,C,Y,X are concyclic. (ii)\text{(ii)} II is also the incentre of triangle DYXDYX.
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