Problem

Source: 2009 Peru Iberoamerican TST problem 3

Tags: geometry, geometry unsolved



Let $M, N, P$ be the midpoints of the sides $AB, BC, CA$ of a triangle $ABC$. Let $X$ be a fixed point inside the triangle $MNP$. The lines $L_1, L_2, L_3$ that pass through point $X$ are such that $L_1$ intersects segment $AB$ at point $C_1$ and segment $AC$ at point $B_2$; $L_2$ intersects segment $BC$ at point $A_1$ and segment $BA$ at point $C_2$; $L_3$ intersects segment $CA$ at point $B_1$ and segment $CB$ at point $A_2$. Indicates how to construct the lines $L_1, L_2, L_3$ in such a way that the sum of the areas of the triangles $A_1A_2X, B_1B_2X$ and $C_1C_2X$ is a minimum.