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diameter of circumcircle bisector of BC, divides AC in 1:3 iff tan B = 2 tan C

Source: 2015 (-16) Swedish Mathematical Competition p1

August 31, 2020
geometrycircumcirclediametertrigonometry

Problem Statement

Given the acute triangle ABCABC. A diameter of the circumscribed circle of the triangle intersects the sides ACAC and BCBC, dividing the side BCBC in half. Show that the same diameter divides the side ACAC in a ratio of 1:31: 3, calculated from AA, if and only if tan⁡B=2tan⁡C\tan B = 2 \tan C.
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