Triangle Centers and Kiepert's Hyperbola

Abstract

In this paper, we discuss the proofs of the primary classical triangle centers and Kiepert's Hyperbola as a solution to Lemoine's Problem. The definitions of terms which will be used throughout the paper are presented. A brief description of well-known triangle centers as well as complete proofs of the remaining classical triangle centers is provided. Many of the proofs of the classical triangle centers require the use of Ceva's Theorem. Ceva's Theorem is proven in the beginning prior to the introduction of the triangle centers. We also explore the proof of Kiepert's Hyperbola as a solution to a problem posed by Lemoine in 1868. A proof of the Nine-Point Circle is provided since the center of Kiepert's Hyperbola lies on the Nine-Point Circle. The trilinear coordinate system provides the basis for the proof of Kiepert's Hyperbola. A brief description of the system and the proofs of its primary theorems are given. The proof of Kiepert's Hyperbola is given along with its properties.