Cubic lines relative to a triangle (Q1114930)

An extension of the traditional geometry of triangle is derived by the author through the construction of two 30-points particular cubics K and K', and a dual correspondence between them is stated. The corresponding tangents to the cubic K at all the 30 points are also significative lines of the triangle. For a proof, a trilinear normal coordinates system of the reference triangle is applied. Finally, a complete summary of all the 30 significative points appearing in the exposition and expressed in the trilinear normal coordinates system is included in the Appendix.