Dujmović, Eppstein, Suderman, and Wood showed that every 3-connected plane graph G with n vertices ad-mits a straight-line drawing with at most 2.5n − 3 seg-ments, which is also the best known upper bound when restricted to plane triangulations. On the other hand, they showed that there exist triangulations requiring 2n − 6 segments. In this paper we show that every plane triangulation admits a straight-line drawing with at most (7n − 2∆0 − 10)/3 ≤ 2.33n segments, where ∆0 is the number of cyclic faces in the minimum re-alizer of G. If the input triangulation is 4-connected, then our algorithm computes a drawing with at most (9n − 9)/4 ≤ 2.25n segments. For general plane graphs with n vertices and m edges, our algorithm requires at most (16n − 3m − 28)/3 ≤ 5.33n −m segments, which is smaller than 2.5n − 3 for all m ≥ 2.84n.