The triple quad formula is a relation invented by Canadian-Australian geometer Norman Wildberger that explains the interaction between the quadrances between any two of three collinear points in the affine -space over an arbitrary field not of characteristic .
Let an invertible matrix define a symmetric bilinear form on , the -dimensional vector space associated to , as follows:
Here, we assume vectors in to be written in row form, i.e. as matrices.
Suppose that with
precisely when are collinear.
Without loss of generality, we work in and let
Suppose the matrix defines a symmetric bilinear form on as defined previously, so that
We then have that
Since the two are equal, our desired result is obtained.
Suppose we are working in -dimensional Euclidean space over the rational number field , and let and be two of the quadrances associated to any two of the three collinear points and . To solve for , we substitute these quantities into the triple quad formula to get
This simplifies to the quadratic equation
Factorising the left side into , we have that or . This agrees with our usual notions of "lengths" in Euclidean space; the two answers take into account that we have not imposed an "ordering" of the points on the line.