# Isotomic Transformation with Respect to a Family of Triangles

## Abstract

The isotomic conjugate of an arbitrary point *P* with respect to a given triangle \(\triangle ABC\) 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 \(\triangle ABC\) 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 \(\triangle ABC_i\), 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.

## Keywords

Isotomic conjugate Family of triangles Cubic transformation Barycentric coordinates## References

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