Isotomic Transformation with Respect to a Family of Triangles

摘要

The isotomic conjugate of an arbitrary point P with respect to a given triangle ?ABC is the intersection point P' of the isotomic lines relative to the cevians through the point P. Furthermore, the isotomic transformation of a geometric object is defined by finding isotomic conjugates with respect to a triangle ?ABC of all points on the geometric object. The isotomic transformation is a quadratic transformation for which various properties are known. In this article the isotomic transformation will be applied to a given fixed point P with respect to a one-parameter family of triangles AABCi, such that two vertices A, B are fixed and the third vertex C_i lies on a geometric object. It will be shown that this new transformation is a cubic transformation and some properties will be stated.