  In this article we study Ceva's theorem and its higher-dimensional extensions
from the perspective of algebraic and projective geometry. First, we situate
the theorem within the study of algebraic surfaces by relating it to the
defining equation of a del Pezzo surface of degree six inside the product of
three projective lines. Second, by interpreting (higher-dimensional analogues
of) Ceva's theorem in terms of projections from projective spaces, we recast
these results as matrix completion problems. We use these ideas to offer proofs
of some higher-dimensional analogues of Ceva's theorem. This article is written
with a nonspecialist audience in mind and we hope that some useful context is
provided in the form of remarks in the sections on surfaces for students of
algebraic geometry.
