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Triangle with an arbitrary point inside

Source: Ukrainian TST 2008 Problem 9

February 12, 2009
geometryparallelogramratiogeometric transformationhomothetygeometry unsolved

Problem Statement

Given △ABC \triangle ABC with point D D inside. Let A_0\equal{}AD\cap BC, B_0\equal{}BD\cap AC, C_0 \equal{}CD\cap AB and A1 A_1, B1 B_1, C1 C_1, A2 A_2, B2 B_2, C2 C_2 are midpoints of BC BC, AC AC, AB AB, AD AD, BD BD, CD CD respectively. Two lines parallel to A1A2 A_1A_2 and C1C2 C_1C_2 and passes through point B0 B_0 intersects B1B2 B_1B_2 in points A3 A_3 and C3 C_3respectively. Prove that \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}.
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