Harmonic Conjugates

The operation of taking the harmonic conjugate with respect to a pair of points preserves the predicates of inversive geometry. For example, in the following diagram, the circle ABCD is sent to the circle A'B'C'D' by taking harmonic conjugates with respect to the points E and F. Furthermore, the points A, B, C, and D form a harmonic quadruple, and the image points A', B', C', and D' also form a harmonic quadruple.

Harmonic image of a circle

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610