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tangent circumcircles if BA_1 = 3A_1C, altitudes, orthocenter related

Source: 2019 Ukraine TST 2.3

November 17, 2020
geometrycircumcircletangent circlesatltitudesorthocenter

Problem Statement

Given an acute triangle ABCABC . It's altitudes AA1,BB1AA_1 , BB_1 and CC1CC_1 intersect at a point HH , the orthocenter of △ABC\vartriangle ABC. Let the lines B1C1B_1C_1 and AA1AA_1 intersect at a point KK, point MM be the midpoint of the segment AHAH. Prove that the circumscribed circle of △MKB1\vartriangle MKB_1 touches the circumscribed circle of △ABC\vartriangle ABC if and only if BA1=3A1CBA1 = 3A1C.
(Bondarenko Mykhailo)
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