Difference between revisions of "Cevian"
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− | A cevian is a line segment that extends from one vertex of a [[triangle]] to the opposite side. [[ | + | ==Definition== |
+ | |||
+ | A '''cevian''' is a [[line segment]] or [[ray]] that extends from one [[vertex]] of a [[polygon]] (usually a triangle) to the opposite side (or the extension of that side). In the below diagram, <math>AD</math> is a cevian. | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(100,0)--(10,50)--(0,0)); | ||
+ | draw((10,50)--(70,0)); | ||
+ | dot((10,50)); | ||
+ | label("$A$",(10,50),N); | ||
+ | dot((0,0)); | ||
+ | label("$B$",(0,0),SW); | ||
+ | dot((100,0)); | ||
+ | label("$C$",(100,0),SE); | ||
+ | dot((70,0)); | ||
+ | label("$D$",(70,0),S); | ||
+ | </asy> | ||
+ | |||
+ | ==Special Cevians== | ||
+ | |||
+ | * A [[triangle median|median]] is a cevian that divides the opposite side into two congruent lengths. | ||
+ | * An [[altitude]] is a cevian that is perpendicular to the opposite side. | ||
+ | * An [[angle bisector]] is a cevian that divides the angle the cevian came from in half. | ||
+ | |||
+ | ==Finding Lengths== | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(100,0)--(10,50)--(0,0)); | ||
+ | draw((10,50)--(70,0)); | ||
+ | dot((10,50)); | ||
+ | label("$A$",(10,50),N); | ||
+ | dot((0,0)); | ||
+ | label("$B$",(0,0),SW); | ||
+ | dot((100,0)); | ||
+ | label("$C$",(100,0),SE); | ||
+ | dot((70,0)); | ||
+ | label("$D$",(70,0),S); | ||
+ | </asy> | ||
+ | |||
+ | In the diagram, note that <math>\cos{ \angle ADB} = -\cos{ \angle ADC}</math> because <math>\angle ADB + \angle ADC = 180^\circ</math>. Thus, | ||
+ | <cmath>\frac{AD^2 + DB^2 - AB^2}{2 \cdot AD \cdot DB} = -\frac{AD^2 + DC^2 - AC^2}{2 \cdot AD \cdot DC}</cmath> | ||
+ | |||
+ | == See also == | ||
+ | * [[Ceva's Theorem]] | ||
+ | * [[Angle Bisector Theorem]] | ||
+ | * [[Stewart's Theorem]] | ||
+ | |||
+ | {{stub}} | ||
+ | |||
+ | [[Category:Definition]] | ||
+ | [[Category:Geometry]] |
Latest revision as of 00:35, 19 June 2018
Contents
[hide]Definition
A cevian is a line segment or ray that extends from one vertex of a polygon (usually a triangle) to the opposite side (or the extension of that side). In the below diagram, is a cevian.
Special Cevians
- A median is a cevian that divides the opposite side into two congruent lengths.
- An altitude is a cevian that is perpendicular to the opposite side.
- An angle bisector is a cevian that divides the angle the cevian came from in half.
Finding Lengths
In the diagram, note that because . Thus,
See also
This article is a stub. Help us out by expanding it.