In this paper we are going to generalise some standard results about the ellipses of least or greatest area, respectively circumscribing or inscribed in a triangle, to the corresponding ellipses for a convex quadrilateral.
Given ΔABC let D, E, F be the midpoints of BC, CA, AB respectively; then the medians AD, BE, CF concur at the centroid G of ΔABC. There is an ellipse with centre G, touching the sides of ΔABC at their respective midpoints. It is known as the midpoint ellipse or Steiner inellipse of ΔABC, and it is the ellipse of greatest area inscribed in ΔABC. Since AG : GD = 2 : 1, and similarly for the other medians, a central dilation (an enlargement) with centre G and scale factor –2 (or, equivalently, 2) applied to the Steiner inellipse produces a similar ellipse, also with centre G, and passing through A, B and C. This is the Steiner circumellipse of ΔABC, and it is the ellipse of least area circumscribing ΔABC. See Figure 1. For more detail, see [1, 2] and the references given there.