An observation on certain point-line configurations in classical planes (Q1199586)

The authors define a cotangency set in \(PG(2,K)\) to be a set \(S\) of points with an injective mapping \(f\) from \(S\) into the set of lines of \(PG(2,K)\) satisfying the following two properties: (1) \(x\notin f(x)\) for every \(x\in S\); (2) given any two distinct points \(x,y\in S\), the point \(f(x)\cap f(y)\) belongs to the line through \(x\) and \(y\). The authors prove that a cotangency set never contains a quadrangle. The proof of this theorem is remarkably short (less than 20 lines). Then they show that a number of known results on sets of points external to a hermitian curve or to a conic are actually trivial corollaries of their theorem.