Where do Euler, Soddy and Gergonne fit in a tetrahedron?
Adrian Oldknow
1. Introduction
In [1] I reported computer-aided discoveries made in triangle geometry, together with
an analysis using trilinear homogeneous coordinates. These started from a simple
problem about three disks touching each other in pairs. With the release of the Cabri
3D software in 2004 I was able to extend the study to seek analogous results for the
geometry of the tetrahedron and to make new discoveries. Working with Michael Fox,
John Rigby and Sir Christopher Zeeman we now have proofs and an analysis for these
results, together with interesting connections with other aspects of geometry such as
Steiner chains. We have submitted a two part article to the Mathematical Gazette which
we hope will be published, maybe later in 2007. In the meantime I am using this note
just to communicate the principal results here, which interested readers can verify for
themselves using Cabri 3D.
2. Review of results for triangles in 2D
It is well known that the circum- and ortho-centres of a triangle are collinear with the
centroid in the Euler line (except in the case of the equilateral triangle). In the triangle
A =ABC let the incircle, centre I, touch the side BC at the point of tangency ZBC etc.
The lines AZBC etc. are concurrent in the Gergonne point G. Thus the tangent (or
Gergonne) triangle Z = ZABZBCZCA is in perspective with A = ABC with centre G. Their
perspective axis is perpendicular to IG and passes through the harmonic conjugates NBC
of ZBC w.r.t. BC etc. I called this axis the Gergonne line, and named the conjugate
points NBC as the Nobbs points.
The vertex circle OB has centre B and passes through ZBC etc. The vertex circles OA, OB,
OC touch each other in pairs externally at the points of tangency. The inner- and outerSoddy circles OS, OS’ have centres S, S and touch each of the three vertex circles. The
line SS is called the Soddy line of the triangle, and passes through I and G. The points
S,S and I,G are in harmonic range.
We denote the points of contact of the Soddy circles with the vertex circle OA as TA and
TA. Thus we have the inner and outer Soddy triangles T = TATBTC and T =TATBTC.
The common tangents at these contact points form inner and outer Soddy tangential
triangles V =VAVBVC and V = VAVBVC. There are 15 pairs of the six triangles A, Z, T,
V, T, V which are each in perspective from one of 10 distinct centres on the Soddy
line and which each have the Gergonne line as axis.
We denote the radius of OA by a, and define its reciprocal (i.e. its curvature or bend) by
 = 1/a. Denoting the radius of the incircle by r, with curvature , and working with
barycentric (or areal) coordinates we can express each of the ten centres in the form
I + G . These, together with their corresponding value of , are:
Gr{-4}, O{-2}, R{-4/3}, S{-1}, I{0}, S{1}, R{4/3}, O{2}, Gr{4}, G{}.
I named the new points after Brian Griffiths (Gr), John Rigby (R) and myself (O).
3. Special tetrahedra in 3D
First we note that while any tetrahedron A = ABCD has a centroid, a circumcentre and
an incentre, it only has an orthocentre under a special condition – namely that opposite
edges, such as AB and CD, are perpendicular. Such special tetrahedra do have an Euler
line, and there is also a generalisation of the 9 point circle to a 12 point sphere.
The equivalent to our three-disk problem is now a `four-ball’ problem. Given any three
balls (spheres) of radii a,b,c it is always possible to arrange them on a plane so that they
touch each other. Their centres A, B, C form a triangle. Provided that the radius d of
the fourth ball is larger than the inner-Soddy radius of ABC then it can be placed on top
of the other three balls, and the four centres now form a tetrahedron A = ABCD. But a
general tetrahedron can be formed from any six segments of arbitrary length – so the 4ball tetrahedra (FBT) are a special class for which the sums of lengths of pairs of
opposite sides are equal e.g. AB+CD = a+b+c+d etc.
4. Review of results for 4-ball tetrahedra in 3D
It would seem natural to think that the insphere would take on the role of the incircle,
but that turns out not to be the case. The pairs of vertex spheres such as OA, OB touch in
tangent points ZAB on the edges, and these are the points of contact of the midsphere (or
intersphere) centre M. The intersection of the midsphere with each triangular face of
the tetrahedron is its incircle, and so the normal to each face through its incentre passes
through M. Barycentric coordinates of key points are A(1,0,0,0), ZAB(, , 0, 0) and
M  (  ), (  ),  (   ), (  )  where  =1/a etc. and  = ½ ( +  +  + )
The three joins of opposite points of contact, such as ZABZCD are coincident in the point
Z (, , , ) - the Zeeman point of A – as are the four joins of each vertex to the
Gergonne point of the opposite face, such as AZBCD – see Fig.1. ZBCD is at (0, , , ).
The line MZ in space now takes on the role of the Soddy line (cf IG in the plane).
The Gergonne tetrahedron Z = ZABCZABDZACDZBCD is in perspective with the four-ball
tetrahedron A from centre Z, and so there is also a perspective plane – which is the
Soddy plane perpendicular to the Soddy line MZ and which contains the six Nobbs
points which are the conjugates of the contact points - e.g. NAB (,-, 0, 0) is conjugate
to ZAB wrt A,B. This plane is the polar plane of Z wrt the midsphere. There are four sets
of three Nobbs points, such as NBC, NCA, NAB which are collinear (in the Fox line lD).
Of course there is a pair of Soddy spheres with centres S, S which each touch all four
vertex spheres – see Fig. 2. Since the contacts need not always be external (or internal)
these are now called the first and second Soddy spheres. Again S,S and M,Z are in
harmonic range. As before, we denote the points of contact of the Soddy spheres with
the vertex sphere OA as TA and TA etc. Thus we have the first and second Soddy
tetrahedra T = TATBTCTD and T =TATBTCTD. The common tangent planes at these
contact points form first and second Soddy tangential tetrahedra V =VAVBVCVD and
V = VAVBVCVD. There are again 15 pairs of the six tetrahedra A, Z, T, V, T, V
which are now each in perspective from one of 8 distinct centres on the Soddy line and
which each have the Soddy plane as axis.
Writing  = 1/a for the curvature (or bend) of the vertex sphere OA, we set  = /23,
where   (      )2  2( 2  2   2  2 ). Again, working with barycentric
coordinates we can express each of the eight centres on the Soddy line in the form
M + kZ. These, together with their corresponding value of k are:
O{-3}, R{-3/2}, S{-1}, M{0}, S{1}, R{3/2}, O{3}, Z{}.
So there are 3D versions of Rigby and Oldknow points, but not of Griffiths points.
5. Conclusion
As is usual with most interesting mathematics, the results are easy to convey but belie
the struggle to establish them. The discoveries would not have been possible without
the aid of the computer software Cabri 3D. Of course a discovery, by itself, just
remains a mathematical conjecture until it can be proved. All the results above have
been proved by analytic means – using barycentric coordinates - which often also
required computer assistance in the form of a computer algebra system such as Derive.
But such algebraic approaches, even when they are as elegant as they turned out to be,
rarely throw up any geometric insights – and so we also have a full set of proofs for our
results based on more obviously geometric techniques such as vectors and inversions.
These reveal interesting links with Steiner chains.
6. References
1
A. Oldknow, The Euler-Soddy-Gergonne triangle of a triangle, American Math
Monthly 103(4) (1996) 319-329.
2
C. Zeeman, Three-dimensional theorems for schools, The Mathematical
Association, Leicester UK, 2005.
Figure 1. The midsphere centre M and the
Zeeman point Z.
Figure 2. Three of the vertex spheres
and the first Soddy sphere.