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Area of M(L) <= Area of M(L*) [ILL 1971]

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January 1, 2011
geometrygeometry proposed

Problem Statement

Let Li, i=1,2,3L_i,\ i=1,2,3, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths li, i=1,2,3l_i,\ i=1,2,3. By Li∗L_i^{\ast} we denote the segment of length lil_i with its midpoint on the midpoint of the corresponding side of the triangle. Let M(L)M(L) be the set of points in the plane whose orthogonal projections on the sides of the triangle are in L1,L2L_1,L_2, and L3L_3, respectively; M(L∗)M(L^{\ast}) is defined correspondingly. Prove that if l1≥l2+l3l_1\ge l_2+l_3, we have that the area of M(L)M(L) is less than or equal to the area of M(L∗)M(L^{\ast}).
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