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concurrency, feets of arc midpoint of circumcircle on 2 triangle sides

Source: 2021 Cono Sur Shortlist G2 https://artofproblemsolving.com/community/c1088686_cono_sur_shortlist__geometry

October 31, 2022
geometryconcurrencyconcurrentarc midpoint

Problem Statement

Let ABCABC be an acute triangle. Define A1A_1 the midpoint of the largest arc BCBC of the circumcircle of ABCABC . Let A2A_2 and A3A_3 be the feet of the perpendiculars from A1A_1 on the lines ABAB and ACAC , respectively. Define B1B_1, B2B_2, B3B_3, C1C_1, C2C_2, and C3C_3 analogously. Show that the lines A2A3A_2A_3, B2B3B_2B_3, C2C3C_2C_3 are concurrent.
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