A general 3-4-5 puzzle (Q1922880)

Let \(PV_i\), \(i=1, \dots, n+1\), be lengths between a point \(P\) and vertices \(V_i\) of a regular simplex in \(\mathbb{R}^n\) of side \(s\). It is a question, what is \(s\)? \textit{O. Bottema} [`Solution to problem 222', Nieuw Arch. Wisk. 3, 124-125 (1985)] gave a determinant method based on distance geometry to find \(s\) when it exists and \textit{H. Eves} [`Solution to problem 187', College Math. J. 13, 278-282 (1982)] has solved the corresponding problem in the plane for regular \(n\)-gons. This problem is called the 3-4-5 puzzle by \textit{S. Rabinowitz} [`Ptolemy's legacy', MathPro Press, Westford, Mass. (1993)], after the well known special case in \(\mathbb{R}^2\) of a point \(P\) of distances 3,4 and 5 from the vertices of an equilateral triangle. The author extends Eves' observation about the point \(P\) and the circumcircle of a solution to the equilateral triangular case to this general case. Geometry is used to anticipate algebraic detail. Geometry is also used to interpret Klamkin's simultaneous triangle inequality [see \textit{M. S. Klamkin}, Math. Mag. 60, No. 4, 236-237 (1987; Zbl 0633.51012)].