ILL 1971 Problem 41

Let $L_i,\ i=1,2,3$ , be line segments on the sides of an equilateral triangle, one segment on each side, with lengths $l_i,\ i=1,2,3$ . By $L_i^{\ast}$ we denote the segment of length $l_i$ with its midpoint on the midpoint of the corresponding side of the triangle. Let $M(L)$ be the set of points in the plane whose orthogonal projections on the sides of the triangle are in $L_1,L_2$ , and $L_3$ , respectively; $M(L^{\ast})$ is defined correspondingly. Prove that if $l_1\ge l_2+l_3$ , we have that the area of $M(L)$ is less than or equal to the area of $M(L^{\ast})$ .