Cardioids and Self-inversive Cubic Polynomials. Issue 4 (21st April 2019)

• Record Type: Journal Article
• Authors: Holland, Finbarr; Smyth, Roger
• Is Part Of: American Mathematical Monthly. Volume 126:Issue 4(2019)
• Page Start: 319
• Page End: 329
• Publication Date: 2019-04-21
• Subjects: MSC: Primary 30C10 -- Secondary 14H45; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2019.1568151
• Languages: English
• ISSNs: 0002-9890

Abstract

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.