The Metamorphosis of the Butterfly Problem

Abstract

Editor's note. This article illustrates the diversity of geometric techniques that can be brought to bear on a single problem. The author was prompted to examine his ample collection of historical material when a compilation of varied proofs of the Butterfly problem was offered by Kaidy Tan of Fukien Teacher's University, Foochow, Fukien, China. All of these proofs had appeared in print, and this article outlines many of them, providing their historical roots. One of the hardiest of the hardy perennials in the realm of Euclidean geometry is the problem that was dubbed The Butterfly by some as yet unidentified poetic mathematician who fancied the image of a lepidopterous creature in the configuration of the problem. This appellation made its first appearance as the title of solutions published in the American Mathematical Monthly in February 1944 [1]. The name took hold and has probably contributed to some extent to the recent popularity of the problem. My love affair with The Butterfly began thirty years ago with the publication of the following proposal in School Science and Mathematics: In a circle (0), P is the midpoint of chord AB. Chords RS and TV pass through the point P. RV cuts AP at a point M, and ST cuts PB at point N. Prove by high school geometry that MP equals PN.