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_|_ wanted, // diameters of tangent circles (2011 Kyiv City MO Round2 9.4 10.4)

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July 31, 2020
geometryperpendicularparalleltangent circles

Problem Statement

Let two circles be externally tangent at point CC, with parallel diameters A1A2,B1B2A_1A_2, B_1B_2 (i.e. the quadrilateral A1B1B2A2A_1B_1B_2A_2 is a trapezoid with bases A1A2A_1A_2 and B1B2B_1B_2 or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines A1B2A_1B_2 and A2B1A_2B_1 as well intersects those lines at points M,NM, N. Prove that the line MNMN is perpendicular to the parallel diameters A1A2,B1B2A_1A_2, B_1B_2.
(Yuri Biletsky)
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