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lines concurrency, related to circle with one end of diameter the orthocenter

Source: Sharygin 2005 finals 10.6

August 29, 2019
geometryconcurrencyconcurrentorthocentercircle

Problem Statement

Let HH be the orthocenter of triangle ABCABC, XX be an arbitrary point. A circle with a diameter of XHXH intersects lines AH,BH,CHAH, BH, CH at points A1,B1,C1A_1, B_1, C_1 for the second time, and lines AXBX,CXAX BX, CX at points A2,B2,C2A_2, B_2, C_2. Prove that lines A1A2,B1B2,C1C2_1A_2, B_1B_2, C_1C_2 intersect at one point.
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