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ASU 205 All Soviet Union MO 1975 a) similar triangles, b) parallelogram

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July 5, 2019
geometrysimilar trianglesparallelogramCircumcentertangential

Problem Statement

a) The triangle ABCABC was turned around the centre of the circumscribed circle by the angle less than 180180 degrees and thus was obtained the triangle A1B1C1A_1B_1C_1. The corresponding segments [AB][AB] and [A1B1][A_1B_1] intersect in the point C2,[BC]C_2, [BC] and [B1C1][B_1C_1] -- A2,[AC]A_2, [AC] and [A1C1][A_1C_1] -- B2B_2. Prove that the triangle A2B2C2A_2B_2C_2 is similar to the triangle ABCABC.
b) The quadrangle ABCDABCD was turned around the centre of the circumscribed circle by the angle less than 180180 degrees and thus was obtained the quadrangle A1B1C1D1A_1B_1C_1D_1. Prove that the points of intersection of the corresponding lines ( (AB(AB) and (A1B1),(BC)(A_1B_1), (BC) and (B1C1),(CD)(B_1C_1), (CD) and (C1D1),(DA)(C_1D_1), (DA) and (D1A1)(D_1A_1) ) are the vertices of the parallelogram.
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