Equimodular Polynomials and the Tritangency Theorems of Euler, Feuerbach, and Guinand

摘要

Abstract We strengthen a result of Lehmer, obtaining a new necessary condition for the roots of a complex polynomial to have equal modulus. From this we derive the famous theorem of Feuerbach, as well as the less well-known theorems of Euler and Guinand on the tritangent centers of a triangle. The latter theorems constrain the possible locations of the incenter and excenters subject to fixed locations for the circumcenter and orthocenter.