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 
.
