Cevian: Difference between revisions
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==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 == | == See also == | ||
Latest revision as of 00:35, 19 June 2018
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.
![[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]](http://latex.artofproblemsolving.com/3/f/d/3fd24eea624f880d12f92f3d0633bf8831c7c057.png)
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,
![\[\frac{AD^2 + DB^2 - AB^2}{2 \cdot AD \cdot DB} = -\frac{AD^2 + DC^2 - AC^2}{2 \cdot AD \cdot DC}\]](http://latex.artofproblemsolving.com/9/b/2/9b2550e4a7beba62dacb7b22ef2ffee013df42f0.png)
See also
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