Distinct values of bilinear forms on algebraic curves

Any p, q ∈ R2 together with the origin determine a pinned triangle. We prove that any finite set S contained in an irreducible algebraic curve C of degree d in R2 determines Ωd(|S| 4/3) distinct pinned triangle areas, unless C is a line, or an ellipse or hyperbola centered at the origin. This improves the bound Ω(|S|5/4) obtained by Charalambides [1]. The proof is based on that of Pach and De Zeeuw [6], who proved a similar statement for Euclidean distances. The main motivation for this paper is that for pinned triangle areas, this approach becomes more natural, and should better lend itself to generalization.