Figure 4: A Steiner Conic

Figure 4: Steiner's Conic

This figure shows three Pappus lines,

P(A,B,C; D,E,F),
P(A,B,C; F,D,E) and
P(A,B,C; E,F,D).

which meet a common point. This point is labeled S_e and is said to be a Steiner point, in honour of Jacob Steiner (1796-1863).

The locus of S_e as a function of F on the line L₂ is a conic which passes through the five points D, E, AE.BD, BE.CD and CE.AD.

One might guess that this locus contains the point AE.BD, since the Pappus line P(A,B,C; D,E,F) contains the point AE.BD and all the Pappus lines sweep out a full revolution as F traverses the line L₂.
A similar argument can be made for the statement that
the locus of S_e includes the points BE.CD and CE.AD
since the Pappus lines P(A,B,C; F,D,E) and P(A,B,C; E,F,D) pivot on BE.CD and CE.AD, respectively.

The Cabri Geometry II source code is available for all figures used in this paper, as well as demonstration versions of Cabri Geometry II for Windows and Macintosh operating systems.