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Problem 6 of Finals

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

January 11, 2020
geometryInscribed circleexcirclecollinearconcurrent

Problem Statement

In ΔABC\Delta ABC points A1A_1, B1B_1, and C1C_1 are the tangential points of the excircles of ABCABC with its sides. a) Prove that AA1AA_1, BB1BB_1, and CC1CC_1 intersect in one point NN. b) If AC+BC=3ABAC+BC=3AB, prove that the center of the inscribed circle of ABCABC, its tangential point with ABAB, and the point NN are collinear.
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