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Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of T . This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.doi:10.4169/amer.math.monthly.121.07.632 fatcat:c4ex4l4o6vhrxnup5wtezjcguy