In this paper, we present concise proofs of several recently developed exact cone-beam reconstruction methods in the Tuy inversion framework, including both filtered-backprojection and backprojection-filtration formulas in the cases of standard spiral, nonstandard spiral, and more general scanning loci. While a similar proof of the Katsevich formula was previously reported, we present a new proof of the Zou and Pan backprojection-filtration formula. Our proof combines both odd and even data extensions so that only the cone-beam transform itself is utilized in the backprojection-filtration inversion. More importantly, our formulation is valid for general smooth scanning curves, in agreement with an earlier paper from our group [Ye, Zhao, Yu, and Wang, Proc. SPIE 5535, 293-300 (Aug. 6 2004)]. As a consequence of that proof, we obtain a new inversion formula, which is in a two-dimensional filtering backprojection format. A possibility for generalization of the Katsevich filtered-backprojection reconstruction method is also discussed from the viewpoint of this framework.