Integer tetrahedra

[John Conway]

   Hi Ed!  Nice to find someone interested in sphenoids
(see below).  John Conway

On Tue, 8 May 2001, Ed Pegg Jr wrote:

> A few days ago, I asked for a tetrahedron with integer
> sides, with an interior point an integer distance from
> all vertices.  I discovered a spectacular result.
>
> A face of a skew tetrahedra has faces {a,b,c}.

    I presume you're referring to the kind of tetrahedron
("tetrahedra" is plural) whose faces are all congruent,
and have edge-lengths  a,b,c ?  [This kind of tetrahedron
is traditionally called a "sphenoid", by the way.]

> What is the radius of the Circumsphere?
>
>    Sqrt[(a^2 + b^2 + c^2)/8]
>
> Is this known? ( It's too nice a result for it to be mine. :-) )

   Sphenoids have been much studied, so more or less everything
about them is "known", though maybe to only a few people.

   The great thing about sphenoids is that they can be obtained by
taking alternate vertices of a rectangular box, or "cuboid", which
usually gives easy proofs.

   In this case, take the vertices to be

     (x,y,z), (x,-y,-z), (-x,y,-z), (-x,-y,z),

so that the squared circumradius is  rr = xx+yy+zz, from which
your formula easily follows using

    aa = 4(yy+zz),  bb = 4(zz+xx),  cc = 4(xx+yy).

   Since the sphenoid is what's left of the box after removing
four "corner" tetrahedra of volume (2x.2y.2z)/6, its volume is
8xyz - 16xyz/3 = 8xyz/3, whose square is  (4xx.4yy.4zz)/9,
easily seen to be

      bb+cc-aa cc+aa-bb aa+bb-cc
      -------- -------- --------  / 9,
         2        2         2

which I think is what you quoted.

    Regards,  JHC