Geometry Theorems for Competitive Math: Exploring Ceva’s Theorem and Triangle Centers

In this video, we introduce Ceva’s Theorem, a powerful tool in geometry for understanding the concurrency of lines in a triangle. We provide two proofs of Ceva’s Theorem—one using triangle areas and another with Menelaus’ Theorem—giving a comprehensive view of how it works. We then apply Ceva’s Theorem to prove three essential geometric facts: the medians, altitudes, and angle bisectors of a triangle are concurrent, leading to the triangle’s centroid, orthocenter, and incenter, respectively. This video is part of the Geometry Theorems for Competitive Math series on Thinking in Math, designed for students preparing for AMC, AIME, and USAMO. In upcoming videos, we’ll dive into the properties of these special centers, starting with the orthocenter, and solve interesting geometry problems that test these concepts.