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How to make the sum of areas a minimum?

Source: 2009 Peru Iberoamerican TST problem 3

May 8, 2023
geometrygeometry unsolved

Problem Statement

Let M,N,PM, N, P be the midpoints of the sides AB,BC,CAAB, BC, CA of a triangle ABCABC. Let XX be a fixed point inside the triangle MNPMNP. The lines L1,L2,L3L_1, L_2, L_3 that pass through point XX are such that L1L_1 intersects segment ABAB at point C1C_1 and segment ACAC at point B2B_2; L2L_2 intersects segment BCBC at point A1A_1 and segment BABA at point C2C_2; L3L_3 intersects segment CACA at point B1B_1 and segment CBCB at point A2A_2. Indicates how to construct the lines L1,L2,L3L_1, L_2, L_3 in such a way that the sum of the areas of the triangles A1A2X,B1B2XA_1A_2X, B_1B_2X and C1C2XC_1C_2X is a minimum.
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