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concurrency related to midpoints and incenters

Source: 1998 Estonia National Olympiad Final Round grade 11 p2

March 11, 2020
geometryincenterconcurrentmidpoints

Problem Statement

In a triangle ABC,A1,B1,C1ABC, A_1,B_1,C_1 are the midpoints of segments BC,CA,AB,A2,B2,C2BC,CA,AB, A_2,B_2,C_2 are the midpoints of segments B1C1,C1A1,A1B1B_1C_1,C_1A_1,A_1B_1, and A3,B3,C3A_3,B_3,C_3 are the incenters of triangles B1AC1,C1BA1,A1CB1B_1AC_1,C_1BA_1,A_1CB_1, respectively. Show that the lines A2A3,B2B3A_2A_3,B_2B_3 and C2C3C_2C_3 are concurrent.
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