# Ceva's Theorem

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

## 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

## Alternate Formulation

The trig version of Ceva's Theorem states that cevians are concurrent if and only if

### Proof

*This proof is incomplete. If you can finish it, please do so. Thanks!*

We will use Ceva's Theorem in the form that was already proven to be true.

First, we show that if , holds true then which gives that the cevians are concurrent by Ceva's Theorem. The Law of Sines tells us that

Likewise, we get

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.