Pick any P. Draw lines from P to each vertex. These lines divide the equilateral triangle into three triangles. Each of these triangles has as its base one side of the equilateral triangle and as its height the perpendicular distance from P to that side. The sum of the areas of these triangles equals the area of the original triangle. So, the sum of the three distances from P to the sides, is the same as the height of the original triangle. Thus, the average distance to the sides is √3/6 times the side of the triangle.
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