Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Incenter"
Line 7: Line 7:
 
== Proof of Existence ==
 
== Proof of Existence ==
  
−
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.  
+
Consider a triangle <math>ABC</math>.  Let <math>I</math> be the intersection of the respective interior angle bisectors of the [[angle]]s <math>BAC</math> and <math>CBA</math>.  We observe that since <math>I</math> lies on an angle bisector of <math>BAC</math>, is equidistant from <math>AB</math> and <math>CA</math>; likewise, it is equidistant from <math>BC</math> and <math>AB</math>; hence it is equidistant from <math>BC</math> and <math>BC</math> and <math>CA</math> and therefore lies on an angle bisector of <math>ACB</math>.  Since it lies within the triangle <math>ABC</math>, this is the interior angle bisector of <math>ACB</math>.  Since <math>I</math> 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]].
 
It should be noted that this proof parallels that for the existance of the [[circumcenter]].
Line 18: Line 18:
  
 
<math>\bullet</math> The unnormalised [[areal coordinates]] of the incenter are <math>(a,b,c)</math>
 
<math>\bullet</math> The unnormalised [[areal coordinates]] of the incenter are <math>(a,b,c)</math>
 +
 +
[[Category:Geometry]]

Revision as of 07:59, 20 October 2007

This article is a stub. Help us out by expanding it.

Triangle ABC with incenter I, with angle bisectors (red), incircle (blue), and inradii (green)

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.

Proof of Existence

Consider a triangle $ABC$. Let $I$ be the intersection of the respective interior angle bisectors of the angles $BAC$ and $CBA$. We observe that since $I$ lies on an angle bisector of $BAC$, is equidistant from $AB$ and $CA$; likewise, it is equidistant from $BC$ and $AB$; hence it is equidistant from $BC$ and $BC$ and $CA$ and therefore lies on an angle bisector of $ACB$. Since it lies within the triangle $ABC$, this is the interior angle bisector of $ACB$. Since $I$ 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.

Properties of the Incenter

$\bullet$ The incenter of any triangle lies within the orthocentroidal circle.

$\bullet$ The unnormalised areal coordinates of the incenter are $(a,b,c)$

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Incenter&oldid=18419"