Cardioids and Self-inversive Cubic Polynomials

Abstract

The standard cardioid is the set of points in the complex plane formed by reflecting the point 1 in every tangent to the unit circle. These points constitute a simple closed curve that is the boundary of two open disjoint regions, a bounded inner region that is heart-shaped, and an outer region. We verify that the outer region consists of all the points from which three tangents can be drawn to the cardioid, a statement that is part of the folklore of the theory of the cardioid—and deemed by many to be geometrically obvious! It was the key observation that led Frank Morley to his celebrated trisector theorem.