Soddy Line

A Line on which the Incenter $I$, Gergonne Point Ge, and inner and outer Soddy Points $S$ and $S'$ lie (the latter two of which are the Equal Detour Point and the Isoperimetric Point). The Soddy line can be given parametrically by

$$I+\lambda {\it Ge},$$

where $\lambda$ is a parameter. It is also given by

$$\sum (f-e)\alpha=0,$$

where cyclic permutations of $d$, $e$, and $f$ are taken and the sum is over Trilinear Coordinates $\alpha$, $\beta$, and $\gamma$.

$\lambda$	Center
$-4$	Outer Griffiths Point ${\it Gr}'$
$-2$	Outer Oldknow Point ${\it Ol}'$
$-{\textstyle{4\over 3}}$	Outer Rigby Point ${\it Ri}'$
$-1$	Outer Soddy Point $S'$
0	Incenter $I$
1	Inner Soddy Point $S$
${\textstyle{4\over 3}}$	Inner Rigby Point Ri
2	Inner Oldknow Point Ol
4	Inner Griffiths Point Gr
$\infty$	Gergonne Point

$S'$, $I$, $S$, and Ge are Collinear and form a Harmonic Range (Vandeghen 1964, Oldknow 1996). There are a total of 22 Harmonic Ranges for sets of four points out of these 10 (Oldknow 1996).

The Soddy line intersects the Euler Line in the de Longchamps Point, and the Gergonne Line in the Fletcher Point.

See also de Longchamps Point, Euler Line, Fletcher Point, Gergonne Point, Griffiths Points, Harmonic Range, Incenter, Oldknow Points, Rigby Points, Soddy Points

References

Oldknow, A. ``The Euler-Gergonne-Soddy Triangle of a Triangle.'' Amer. Math. Monthly 103, 319-329, 1996.

Vandeghen, A. ``Soddy's Circles and the De Longchamps Point of a Triangle.'' Amer. Math. Monthly 71, 176-179, 1964.