In , W. D. Munn proved the following result.
Theorem 1: Infinitely many ellipses pass through the four vertices of a given convex quadrilateral.
Much of the geometry that I studied as an undergraduate in the 1950s concerned complex projective space, in which convexity plays no part. So I found Theorem 1 especially piquant and sought to understand it better. This article is the result. After examining the convexity of quadrilaterals in general, especially those inscribed in conics, I consider the following problem. Let P be a variable point in the plane, distinct from the vertices of a given convex quadrilateral ABCD. It is well known that there is a unique conic, S (P), through the five points A, B, C, D and P. How does the nature of this conic depend on the position of P? As a spin-off, we get a very short proof of Theorem 1. Finally I look at what happens when the quadrilateral ABCD is not convex. In this case, S (P) is always a hyperbola, but the distribution of A, B, C and D on its branches is still of interest.