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concurrency by joining midpoints of open polygonals XYZ (midpoint of XY+YZ)

Source: Rioplatense Olympiad 1998 level 3 P5

September 10, 2018
geometrymidpointsconcurrencyconcurrent

Problem Statement

We say that MM is the midpoint of the open polygonal XYZXYZ, formed by the segments XY,YZXY, YZ, if MM belongs to the polygonal and divides its length by half. Let ABCABC be a acute triangle with orthocenter HH. Let A1,B1,C1,A2,B2,C2A_1, B_1,C_1,A_2, B_2,C_2 be the midpoints of the open polygonal CAB,ABC,BCA,BHC,CHA,AHBCAB, ABC, BCA, BHC, CHA, AHB, respectively. Show that the lines A1A2,B1B2A_1 A_2, B_1B_2 and C1C2C_1C_2 are concurrent.
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