This article compares different proving methods for projective incidence theorems. In particular, a technique using quadrilateral tilings recently introduced by Sergey Fomin and Pavlo Pylyavskyy is shown to be at most as strong as proofs using bi-quadratic final polynomials and thus, also proofs using Ceva-Menelaus-tilings. Furthermore, we demonstrate the equivalence between quadrilateral-tiling-proofs and proofs using exclusively Menelaus configurations. We exemplify the transition between the proofs in several examples in 2D and in 3D.

翻译：本文比较了射影几何中不同证明方法对于关联定理的适用性。特别地，我们证明了由Sergey Fomin和Pavlo Pylyavskyy最近提出的四边形平铺证明技术，其证明能力至多与使用双二次终多项式的方法相当，因此也不超过使用塞瓦-梅涅劳斯平铺的证明方法。此外，我们证明了四边形平铺证明与仅使用梅涅劳斯构型的证明方法具有等价性。我们通过多个二维和三维实例，具体展示了这些证明方法之间的转换过程。