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Midpoints of segments forming triangles with equal areas.

Source: Vietnam MO 1980 P3

March 17, 2011
geometryanalytic geometry

Problem Statement

Let PP be a point inside a triangle A1A2A3A_1A_2A_3. For i=1,2,3i = 1, 2, 3, line PAiPA_i intersects the side opposite to AiA_i at BiB_i. Let CiC_i and DiD_i be the midpoints of AiBiA_iB_i and PBiPB_i, respectively. Prove that the areas of the triangles C1C2C3C_1C_2C_3 and D1D2D3D_1D_2D_3 are equal.
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