Conjugacies in the plane of a triangle

Summary.

In the literature of triangle geometry, there are several kinds of conjugacies. The purpose of this paper is to recognize and generalize the functional forms for these conjugacies as represented in homogeneous coordinates. We introduce the U-Hirst inverse of X, which is a conjugacy given for points U = u : v : w and X = x : y : z by $U \diamond X = vwx^2 - yzu^2:wuy^2 - zxv^2 : uvz^2 - xyw^2$. Since $U \diamond X = X \diamond U$, the operation $\diamond$ is commutative. Another such case is isoconjugacy, given by $U\,cX = vwyz\,:\,wuzx\,:\,uvxy$ and exemplified by the well-known isogonal and isotomic conjugacies. Others, such as line conjugacies and Ceva conjugacies, are represented by noncommutative operations. Diamond and line conjugacies are members of a two-parameter family of online conjugacies, whereas isoconjugacy and Ceva conjugacy are members of a two-parameter family of offline conjugacies. To the extent that these conjugacies are projective involutions, the underlying projective theory is well known.