The Lemma of Tangents reformulated

Abstract

The famous Lemma of Tangents describes a useful (algebraic) relation between the tangents through three points of an arc in a Desarguesian projective plane. Because the formulation of the lemma assumes the three points to have coordinates ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) and ( 0 , 0 , 1 ) , it is sometimes not so evident to apply when studying arc subsets of more than three points. In this paper, we reformulate the Lemma of Tangents in a concise way which is independent of the chosen basis of the projective plane. We also express the consequences of this lemma for sets of more than three arc points in the form of linear equations. To show that our framework is helpful we provide a new and direct proof of the fact that every q -arc in PG ( 2 , q ) must be part of a conic when q is odd.