Circumradius of a tetrahedron

Question

I put the following formula for the circumradius of a tetrahedron on the Wikipedia page on the tetrahedron, but it was deleted for lack of a citation. Does anyone have a reference for it?

Here, $a,b,c$ are three edges that meet at a point; $A,B,C$ are the opposite edges; and $V$ is the volume of the tetrahedron. $$R = \frac{\sqrt{(aA+bB+cC)(aA+bB-cC)(aA-bB+cC)(-aA+bB+cC)}}{24V}$$

Answer 1

This is a result by A. L. Crelle. It first? appeared in

• Crelle, A. L. "Einige Bemerkungen über die dreiseitige Pyramide." Sammlung mathematischer Aufsätze u. Bemerkungen 1, 105-132, 1821.
  (an online copy is available on archive.org)

The statement appears at page 117 (item 186) and the proof is one or two pages around that. Since the article is in German and I don't speak German, I don't know where the proof exactly starts.

Another English reference is

• I. Todhunter, "Spherical Trigonometry: For the Use of Colleges and Schools" (1886)
  (online copies are available under Project Gutenberg 1 and archive.org 2, 3, 4 ).

The material is covered at page 129 ( Art. 163 ).

Answer 2

In "An Open Problem on Metric Invariants of Tetrahedra" (citation below), Lu Yang and Zhenbing Zeng mention matter-of-factly that the circumradius $R$ satisfies

$$R^2 = -\frac{M_5}{2 M_0}$$

where $M_0$ is the Cayley-Menger determinant, and where $M_5$ is the principal minor determinant obtained by deleting the $5$th row and column of $M_0$. That is, in the notation of OP, $$\begin{align} M_0 := \left|\begin{array}{ccccc} 0 & a^2 & b^2 & c^2 & 1 \\ a^2 & 0 & C^2 & B^2 & 1 \\ b^2 & C^2 & 0 & A^2 & 1 \\ c^2 & B^2 & A^2 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{array}\right| &= 288 V^2 \\[8pt] M_5 := \left|\begin{array}{cccc} 0 & a^2 & b^2 & c^2 \\ a^2 & 0 & C^2 & B^2 \\ b^2 & C^2 & 0 & A^2 \\ c^2 & B^2 & A^2 & 0 \\ \end{array}\right| &= \begin{array}{c} -(a A + b B + c C)(-a A + b B + c C) \\ \cdot(a A - b B + c C)(a A + b B - c C) \end{array} \end{align}$$ which gives the relation OP mentions.

I suspect that the relation should appear in a comprehensive reference of Cayley-Menger lore, but my cursory web search has been fruitless.

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Here's the citation information I have for the Yang-Zeng note:

Lu Yang and Zhenbing Zeng, An open problem on metric invariants of tetrahedra, Proceedings of the 2005 international symposium on Symbolic and algebraic computation (ISSAC ’05). ACM, New York, NY, USA, 362-364. DOI=10.1145/1073884.1073934 http://doi.acm.org/10.1145/1073884.1073934

A PDF of the paper containing the note appears to be available from ResearchGate: https://www.researchgate.net/publication/221563984_An_open_problem_on_metric_invariants_of_tetrahedra