## Isotomic of circumcircle

Here I gather some elementary facts around the isotomic image of the circumcircle of a triangle ABC.
[1] The isotomic image of the circumcircle is the orthic axis L of the anticomplementary A'B'C' of ABC. This line coincides with the trilinear polar of X76 (see IsotomicConicOfLine.html ).
[2] This line is orthogonal to the Euler line and passes through X325 which is also on the tripolar of X99 (Steiner point) of ABC. (The figure below shows some further coincidences of triangle centers on these lines. Primes denote triangle centers of the anticomplementary.)
[3] The isotomic conjugate of the line at infinity is the outer Steiner ellipse of the triangle.
[4] The isotomic conjugate of a line L is an (ellipse/parabola/hyperbola) if and only if the number of intersection points of L with the Steiner outer ellipse is (0/1/2).
[5] The isotomic conjugate of the X99-tripolar is the Kiepert rectangular hyperbola having this line as tangent at the centroid G.

[1] That the circumcircle is the isotomic image of the trilinear polar of X76 is proved in IsotomicConicOfLine.html . In this reference was noticed also that the tripols of all lines passing through the symmedian point K of ABC are points on the circumcircle. The identification of this line with the orthic axis of the anticomplementary is an easy calculation in trilinears. Similar calculations prove [2].
To show [3] use the standard projectivity F mapping the equilateral triangle t0 and its center to t1 = ABC and G correspondingly. This projective map preserves the line at infinity (maps it onto itself) thus induces an affine transformation on the euclidean plane denoted by the same letter F. Now isotomic conjugacy I0 with respect to t0 and isotomic conjugacy I1 with respect to t1 make a commutative diagram with F:

In the case of the equilateral triangle t0 the isotomy coincides with isogonality and using the results of IsogonalOfCircumcircle.html we see that the isotomic with respect to the equilateral of the line at infinity is the circumcircle of t0. This maps under F to the outer Steiner ellipse (see Steiner_Ellipse.html ) and the commutativity above implies that this ellipse is the image of the line at infinity with respect to the isotomy I1.
[4] Is a consequence of [3] and [5] is a standard calculation in trilinears. Note in this respect that the symmedian point K' of the anticomplementary is X69 and is isotomic conjugate to the orthocenter H of ABC. This implies immediately that the isotomic image of line GK' is a rectangular hyperbola.

[1] That the circumcircle is the isotomic image of the trilinear polar of X76 is proved in IsotomicConicOfLine.html . The identification of this line with the orthic axis of the anticomplementary is an easy calculation in trilinears. Similar calculations prove [2].
To show [3] use the standard projectivity F mapping the equilateral triangle t0 and its center to t1 = ABC and G correspondingly. This projective map preserves the line at infinity (maps it onto itself) thus induces an affine transformation on the euclidean plane denoted by the same letter F. Now isotomic conjugacy I0 with respect to t0 and isotomic conjugacy I1 with respect to t1 make a commutative diagram with F:
[3] The isotomic conjugate of the line at infinity is the outer Steiner ellipse of the triangle.
[4] The isotomic conjugate of a line L is an (ellipse/parabola/hyperbola) if and only if the number of intersection points of L with the Steiner outer ellipse is (0/1/2).
[5] The isotomic conjugate of the X99-tripolar is the Kiepert rectangular hyperbola having this line as tangent at the centroid G.