IsotomicOfCircle

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.

Isotomic of circumcircle

See Also