Difference between revisions of "Incenter"
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| + | == Proof of Existence == | ||
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| + | Consider a triangle <math>\displaystyle ABC</math>. Let <math>\displaystyle I</math> be the intersection of the respective interior angle bisectors of the [[angle]]s <math>\displaystyle BAC</math> and <math>\displaystyle CBA</math>. We observe that since <math>\displaystyle I</math> lies on an angle bisector of <math>\displaystyle BAC</math>, is equidistant from <math>\displaystyle AB</math> and <math>\displaystyle CA</math>; likewise, it is equidistant from <math>\displaystyle BC</math> and <math>\displaystyle AB</math>; hence it is equidistant from <math>\displaystyle BC</math> and <math>\displaystyle BC</math> and <math>\displaystyle CA</math> and therefore lies on an angle bisector of <math>\displaystyle ACB</math>. Since it lies within the triangle <math>\displaystyle ABC</math>, this is the interior angle bisector of <math>\displaystyle ACB</math>. Since <math>\displaystyle I</math> is equidistant from all three sides of the triangle, it is the incenter. | ||
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| + | It should be noted that this proof parallels that for the existance of the [[circumcenter]]. | ||
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| + | The proofs of existance for the [[excenter]]s is the same, except that certain angle bisectors are exterior. | ||
Revision as of 19:36, 25 August 2006
This article is a stub. Help us out by expanding it.
The incenter of a triangle is the intersection of its (interior) angle bisectors. The incenter is the center of the incircle. Every nondegenerate triangle has a unique incenter.
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Proof of Existence
Consider a triangle 
. Let 
be the intersection of the respective interior angle bisectors of the angles 
and 
. We observe that since lies on an angle bisector of , is equidistant from 
and 
; likewise, it is equidistant from 
and ; hence it is equidistant from and and and therefore lies on an angle bisector of 
. Since it lies within the triangle , this is the interior angle bisector of . Since is equidistant from all three sides of the triangle, it is the incenter.
It should be noted that this proof parallels that for the existance of the circumcenter.
The proofs of existance for the excenters is the same, except that certain angle bisectors are exterior.
