Concurrency Theorems

An introduction to Ceva's and Menelaus' theorems — powerful criteria for when cevians are concurrent or when points are collinear.

Keywords: concurrency, cevian, Ceva's theorem, Menelaus' theorem, collinearity

Prerequisites: centroid-proof · Difficulty: intermediate

Notes

A cevian is a line segment from a vertex of a triangle to a point on the opposite side. The three medians, altitudes, and angle bisectors are all cevians — and each set meets at a single point (is concurrent).

Why Do They Meet?

Is there a general criterion for when three cevians are concurrent? Ceva's theorem provides exactly this:

Three cevians $AD$, $BE$, $CF$ are concurrent if and only if:

Collinearity

The dual result, Menelaus' theorem, gives a criterion for when three points on the sides (or extensions) of a triangle are collinear.

These two theorems are among the most powerful tools in triangle geometry.

Summary

Ceva's theorem states that cevians $AD$, $BE$, $CF$ are concurrent if and only if $\frac{BD}{DC}\cdot \frac{CE}{EA}\cdot \frac{AF}{FB}=1$.

Menelaus' theorem provides the dual criterion for collinearity on triangle sides and extensions.