Vector Identities Related to the Newton–Gauss Line and Pappus’s Theorem

In plane Euclidean geometry, some classical results can be expressed through closely related algebraic identities. This paper considers a class of identities based on vector cross products and illustrates their use in several geometric configurations. Using the Newton–Gauss line theorem as a representative example, we present two proofs in vector cross product form, one obtained via the method of undetermined coefficients. From the resulting identity, several special cases follow, including The American Mathematical Monthly Problem E1216. The same type of identity is then applied to configurations related to Pappus’s theorem, clarifying how certain algebraic identities correspond to specific area relations, and yielding an additional collinearity property involving centroids.