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concurrent wanted; arc midpoints who are also segment midpoints

Source: 2006 VMEO III Seniors 10.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

September 17, 2021
geometryconcurrencyconcurrentarc midpoint

Problem Statement

Let ABCABC be a triangle inscribed in a circle with center OO. Let A1A_1 be a point on arc BCBC that does not contain A A such that the line perpendicular to OAOA at A1A_1 intersects the lines ABAB and ACAC at two points and the line segment joining those two points has as midpoint A1A_1. Points B1B_1, C1C_1 are determined similarly. Prove that the lines AA1AA_1, BB1BB_1, CC1CC_1 are concurrent.
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