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Concurrency of simso...

Source: Brazil National Olympiad Junior 2021 #5

February 14, 2022
geometry

Problem Statement

Let ABCABC be an acute-angled triangle. Let A1A_1 be the midpoint of the arc BCBC which contain the point AA. Let A2A_2 and A3A_3 be the foot(s) of the perpendicular(s) of the point A1A_1 to the lines ABAB and ACAC, respectively. Define B2,B3,C2,C3B_2,B_3,C_2,C_3 analogously. a) Prove that the line A2A3A_2A_3 cuts BCBC in the midpoint. b) Prove that the lines A2A3,B2B3A_2A_3,B_2B_3 and C2C3C_2C_3 are concurrents.
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