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excenters and projections lead to concurrent lines

Source: 2016 Oral Moscow Geometry Olympiad grades 8-9 p5

July 30, 2019
geometryconcurrencyexcenterexcentersconcurrent

Problem Statement

Points IA,IB,ICI_A, I_B, I_C are the centers of the excircles of ABCABC related to sides BC,ACBC, AC and ABAB respectively. Perpendicular from IAI_A to ACAC intersects the perpendicular from IBI_B to BCB_C at point XCX_C. The points XAX_A and XBX_B. Prove that the lines IAXA,IBXBI_AX_A, I_BX_B and ICXCI_CX_C intersect at the same point.
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