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Max of min of cevian segments

Source: Sharygin CR 2019 P12 (Grade 8 - 11)

March 6, 2019
geometry

Problem Statement

Let A1A2A3A_1A_2A_3 be an acute-angled triangle inscribed into a unit circle centered at OO. The cevians from AiA_i passing through OO meet the opposite sides at points BiB_i (i=1,2,3)(i = 1, 2, 3) respectively.
[*] Find the minimal possible length of the longest of three segments BiOB_iO. [*] Find the maximal possible length of the shortest of three segments BiOB_iO.
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