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concurrency from Chile, circumcircles related (2006 L2 p6)

Source:

May 30, 2019
geometryconcurrentcircumcircle

Problem Statement

Let △ABC \vartriangle ABC be an acute triangle and scalene, with BC BC its smallest side. Let P,Q P, Q points on AB,AC AB, AC respectively, such that BQ=CP=BC BQ = CP = BC . Let O1,O2 O_1, O_2 be the centers of the circles circumscribed to △AQB,△APC \vartriangle AQB, \vartriangle APC , respectively. Sean H,O H, O the orthocenter and circumcenter of △ABC \vartriangle ABC a) Show that O1O2=BC O_1O_2 = BC . b) Show that BO2,CO1 BO_2, CO_1 and HO HO are concurrent
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