Isotomy_OnMedian

1. Isotomy (on median)

Initially defined with respect to some median (AM say) of a triangle ABC. It is the affine transformation, which to each point P corresponds point P', so that PP' is parallel to the base BC of the triangle and the middle N of PP' is on the median AM.
The expression of the transformation in trilinear coordinates is given by:
F_A(x,y,z) = (x',y',z') = (x*sin(B)*sin(C), z*sin(C)², y*sin(B)²).
This is not in absolute trilinears. To obtain the expression in absolute trilinears divide the tripple of coordinates with the factor (sin(B)*sin(C)) of its first coordinate.

2. Symmetry of two parabolas

The figure below shows the two parabolas circumcscribed/inscribed in the triangle ABC and having their axis parallel to the median AM. The corresponding isotomy is an affine symmetry of the two parabolas.
By the way find the five remarkable points on each parabola which can define it as a conic through five points.

3. Distinguish from isotomic conjugation

This kind of affine mapping is to be distinguished from the isotomic conjugation with respect to a triangle, which is a non-affine map and maps lines to conics passing through the vertices of the triangle. Later map is examined in BarycentricCoordinates3.html .

1. Isotomy (on median)

2. Symmetry of two parabolas

3. Distinguish from isotomic conjugation

See Also