An identity for symmetric bilinear forms (Q1061196)

The main result is the following Proposition 2: Let V be a 2-dimensional vector space over a field of characteristic \(\neq 2\) with a symmetric bilinear form f on \(V\times V\). Let \(a_ 0\), \(a_ 1\), \(a_ 2\), \(a_ 3\) be four points on \(C:=\{a\in V| f(a,a)=1\}\) such that \(f(a_ 1,a_ 2)\neq 1\), \(f(a_ 2,a_ 3)\neq 1\) and \(f(a_ 3,a_ 1)\neq 1\). Let \(b_ 1\), \(b_ 2\), \(b_ 3\) be three points in V such that each of the triples \(\{b_ 1,a_ 2,a_ 3\}\), \(\{b_ 2,a_ 3,a_ 1\}\) and \(\{b_ 3,a_ 1,a_ 2\}\) is collinear. If \(f(a_ 0-b_ 1,a_ 2-a_ 3)=0\), \(f(a_ 0-b_ 2,a_ 3-a_ 1)=0\), \(f(a_ 0-b_ 3,a_ 1-a_ 2)=0\) then \(b_ 1\), \(b_ 2\), \(b_ 3\) are collinear. This proposition generalizes a corresponding result of William Wallace (1797) in the case of the real plane \(V={\mathbb{R}}^ 2\). The proof of the proposition needs a certain identity for symmetric bilinear forms (Proposition 1).