# Ceva's Theorem

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

## Contents

## Statement

http://billydorminy.homelinux.com/aopswiki/cevathm.png

A necessary and sufficient condition for AD, BE, CF, where D, E, and F are points of the respective side lines BC, CA, AB of a triangle ABC, to be concurrent is that

where all segments in the formula are directed segments.

## Proof

Let be points on respectively such that are concurrent, and let be the point where , and meet. Draw a parallel to through the point . Extend until it intersects the parallel at a point . Construct in a similar way extending .

*(ceva1.png)*

The triangles and are similar, and so are and . Then the following equalities hold:

and thus

Notice that if directed segments are being used, then and have opposite signs, and therefore when cancelled change the sign of the expression. That's why we changed to .

Now we turn to consider the following similarities: and . From them we get the equalities

which lead to

Multiplying the last expression with (1) gives

and we conclude the proof.

To prove the converse, suppose that are points on respectively and satisfying

Let be the intersection point of with , and let be the intersection of with . Since then are concurrent, we have

and thus

which implies , and therefore are concurrent.

*(proof courtesy planetmath.org, used under GNU License)*

## Example

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 .