Geometry of projective plane and Poisson structure

V.I. Arnold [V. I. Arnold, Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane,Journal of Geometry and Physics. 53 (4) (2005), 421-427] gave an alternative proof to the Lobachevsky triangle altitudes theorem by using a Poisson bracket for quadratic forms and its Jacobi identity, and showed that the orthocenter theorem can be extended on RP2. In this paper, we find a new identity in the Poisson algebra of quadratic forms. Following Arnold's idea, the goal of this article is to give alternative proofs to theorems, of Desargues, Pascal, and Brianchon, in RP2, by using the Poisson bracket and the identity. (C) 2009 Elsevier B.V. All rights reserved.