We study the envelope of the family of lines which bisect the interior region of a
simple, closed curve in the plane. We determine this ‘bisection envelope’for polygons
and show that polygons with no parallel pairs of sides are characterized by their
bisection envelope. We show that the bisection envelope always has at least
three and an odd number of cusps. We investigate the winding numbers of
bisection envelopes, and use this to show that there are an infinite number of
curves with any given bisection envelope and show how to generate them.
We obtain results on the intersections of bisecting lines. Finally, we give a
relationship between the ‘internal area’of a curve and that of its bisection
envelope.
