Two triads of squares

Abstract:

The thirteen points (0,0), $( \pm {a_i},0)$, $i = 1,2,3$, $(0, \pm {b_j})$, $j = 1,2,3$, will be at integer distances from one another if the two triads $a_1^2$, $a_2^2$, $a_3^2$, $b_1^2$, $b_2^2$, $b_3^2$ are such that the nine sums $a_i^2 + b_j^2$ are all perfect squares. Infinite families of solutions are derived from solutions of ${\{ m,n\} ^2} = \{ p,q\} \{ r,s\}$, where $\{ m,n\} = ({m^2} - {n^2})/2mn$, etc. Additional numerical examples are given. Two solutions are given in which one of the triads is extended to a tetrad.