# Ceva's Theorem

**Ceva's Theorem** is an algebraic statement regarding the lengths of cevians in a triangle.

## Contents

## Statement

A necessary and sufficient condition for where and are points of the respective side lines of a triangle , to be concurrent is that

where all segments in the formula are directed segments.

## Proof

Letting the altitude from to have length we have and where the brackets represent area. Thus . In the same manner, we find that . Thus

Likewise, we find that

Thus

## Examples

- Suppose AB, AC, and BC have lengths 13, 14, and 15. If and . Find BD and DC.

If and , then , and . From this, we find and . - See the proof of the concurrency of the altitudes of a triangle at the orthocenter.
- See the proof of the concurrency of the perpendicual bisectors of a triangle at the circumcenter.