Conjugation 2: Conjugate lines in a triangle

摘要

In a previous note [1] the present author has given an account of theconjugations known as isogonal conjugation and isotomic conjugation withrespect to a given triangle and has shown how to generalise the idea ofconjugation involving a pair of points so that P (1, m, n) is q-conjugate, bymeans of a construction involving a transversal q, to the point(p2 1 1, q2I m, r2 / n) and he has also demonstrated how to perform thegeometrical constructions involved. When there is a self-conjugate point at theincentre I one has isogonal conjugation, and when there is a self-conjugatepoint at the centroid G one has isotomic conjugation. This type of conjugationincludes cases in which no self-conjugate point Q exists, this being the casewhen the conjugation is generated by a transversal q that intersects two sidesof the fundamental triangle internally. The purpose of [1] was first to showhow the idea of isogonal and isotomic conjugation can be extended into awhole family of point conjugations of a particular type and secondly to serveas a preparation for extending those ideas to give a consistent definition ofconjugation involving pairs of lines. It is not to be expected that we would beable to announce new results at this stage about the conjugation of points, asthis topic has been studied extensively for over a hundred years. The excuse,if one is needed, for introducing the idea of line conjugations is that by doingso we are able to announce some new results of significance.