Problem #43

You are given an equilateral triangle. The family of all lines that divide the triangle into two pieces of equal area traces out a "deltoid" shape (technically known as the envelope of the lines, the curve that has the family of lines as its tangent lines) as shown in the figure below. The second figure shows the deltoid with the generating lines removed. Find the ratio of the area of the deltoid to the area of the triangle.

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