# Difference between revisions of "Ceva's Theorem"

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== Example == | == Example == | ||

− | Suppose AB, AC, and BC have lengths 13, 14, and 15. If AF | + | Suppose AB, AC, and BC have lengths 13, 14, and 15. If <math>\frac{AF}{FB} = \frac{2}{5}</math> and <math>\frac{CE}{EA} = \frac{5}{8}</math>. Find BD and DC.<br> |

+ | <br> | ||

+ | If <math>BD = x</math> and <math>DC = y</math>, then <math>10x = 40y</math>, and <math>{x + y = 15}</math>. From this, we find <math>x = 12</math> and <math>y = 3</math>. | ||

== See also == | == See also == | ||

* [[Menelaus' Theorem]] | * [[Menelaus' Theorem]] | ||

* [[Stewart's Theorem]] | * [[Stewart's Theorem]] |

## Revision as of 15:10, 20 June 2006

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

## Statement

*(awaiting image)*

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.

## 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 .