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ASU 145 All Soviet Union MO 1971 n parallelograms on a convex n-gon

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July 3, 2019
geometryparallelogramconvex polygon

Problem Statement

a) Given a triangle A1A2A3A_1A_2A_3 and the points B1B_1 and D2D_2 on the side [A1A2][A_1A_2], B2B_2 and D3D_3 on the side [A2A3][A_2A3], B3B_3 and D1D_1 on the side [A3A1][A_3A_1]. If you construct parallelograms A1B1C1D1A_1B_1C_1D_1, A2B2C2D2A_2B_2C_2D_2 and A3B3C3D3A_3B_3C_3D_3, the lines (A1C1)(A_1C_1), (A2C2)(A_2C_2) and (A3C3)(A_3C_3), will cross in one point OO. Prove that if ∣A1B1∣=∣A2D2∣   and   ∣A2B2∣=∣A3D3∣|A_1B_1| = |A_2D_2| \,\,\, and \,\,\, |A_2B_2| = |A_3D_3| then ∣A3B3∣=∣A1D1∣|A_3B_3| = |A_1D_1|
b) Given a convex polygon A1A2...AnA_1A_2 ... A_n and the points B1B_1 and D2D_2 on the side [A1A2][A_1A_2], B2B_2 and D3D_3 on the side [A2A3][A_2A_3], ... BnB_n and D1D_1 on the side [AnA1][A_nA_1]. If you construct parallelograms A1B1C1D1A_1B_1C_1D_1, A2B2C2D2A_2B_2C_2D_2, ...... , AnBnCnDnA_nB_nC_nD_n, the lines (A1C1)(A_1C_1), (A2C2)(A_2C_2), ......, (AnCn)(A_nC_n), will cross in one point OO. Prove that ∣A1B1∣⋅∣A2B2∣⋅...⋅∣AnBn∣=∣A1D1∣⋅∣A2D2∣⋅...⋅∣AnDn∣|A_1B_1| \cdot |A_2B_2|\cdot ... \cdot |A_nB_n| = |A_1D_1|\cdot |A_2D_2|\cdot ...\cdot |A_nD_n|
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