Ceva’s Theorem Is More Than a Formula for Concurrency

Submitted by Kevin rain on

In triangle geometry, one of the most efficient ways to test whether three cevians are concurrent is Ceva’s Theorem.

Let D, E, F lie on sides BC, CA, AB respectively of triangle ABC. Then lines AD, BE, and CF are concurrent if and only if

(BD/DC) * (CE/EA) * (AF/FB) = 1.

What I find interesting is that many students learn this as a “contest trick,” but it is actually a very natural statement. The theorem says that concurrency is encoded by a balance condition on the three side partitions.