# 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 $\sin BAD \sin ACF \sin CBE = \sin DAC \sin FCB \sin EBA$, holds true then $BD\cdot CE\cdot AF = DC \cdot EA \cdot FB$ which gives that the cevians are concurrent by Ceva's Theorem. The Law of Sines tells us that

Likewise, we get

$\sin ACF = \frac{AF}{AC\sin CFA}$ |

$\sin CBE = \frac{CE}{BC\sin BEC}$ |

$\sin CAD = \frac{CD}{AC\sin ADC}$ |

$\sin BCF = \frac{BF}{BC\sin BFC}$ |

$\sin ABE = \frac{AE}{AB\sin AEB}$ |

Thus

$\frac{BD}{AB\sin ADB} \cdot \frac{AF}{AC\sin CFA} \cdot \frac{CE}{BC\sin BEC} = \frac{CD}{AC\sin ADC} \cdot \frac{BF}{BC\sin BFC} \cdot \frac{AE}{AB\sin AEB} |

## 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.