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Ex-centers, middle points, tangencies, concurrent segments

Source: Problem 2, Brazilian MO, 2012

December 8, 2012
geometryincentercircumcirclegeometric transformationhomothetyratiogeometry unsolved

Problem Statement

ABCABC is a non-isosceles triangle. TAT_A is the tangency point of incircle of ABCABC in the side BCBC (define TBT_B,TCT_C analogously). IAI_A is the ex-center relative to the side BC (define IBI_B,ICI_C analogously). XAX_A is the mid-point of IBICI_BI_C (define XBX_B,XCX_C analogously).
Show that XATAX_AT_A,XBTBX_BT_B,XCTCX_CT_C meet in a common point, colinear with the incenter and circumcenter of ABCABC.
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