Editing Incircle and excircles (section)

{{short description|Circles tangent to all three sides of a triangle}}
{{Redirect|Incircle|incircles of non-triangle polygons|Tangential quadrilateral|and|Tangential polygon}}
{{distinguish|Circumcircle}}
[[File:Incircle and Excircles.svg|right|thumb|300px|'''Incircle and excircles of a triangle.'''
{{legend-line|solid black|[[Extended side]]s of triangle {{math|△''ABC''}}}}
{{legend-line|solid #728fce|Incircle ([[incenter]] at {{mvar|I}})}}
{{legend-line|solid orange|Excircles (excenters at {{mvar|J{{sub|A}}}}, {{mvar|J{{sub|B}}}}, {{mvar|J{{sub|C}}}})}}
{{legend-line|solid red|Internal [[angle bisector]]s}}
{{legend-line|solid #32cd32|External angle bisectors (forming the excentral triangle)}}
]]

In [[geometry]], the '''incircle''' or '''inscribed circle''' of a [[triangle]] is the largest [[circle]] that can be contained in the triangle; it touches (is [[tangent]] to) the three sides. The center of the incircle is a [[triangle center]] called the triangle's [[incenter]].<ref>{{harvtxt|Kay|1969|p=140}}</ref>

An '''excircle''' or '''escribed circle'''<ref name="Altshiller-Court 1925 74">{{harvtxt|Altshiller-Court|1925|p=74}}</ref> of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the [[extended side|extensions of the other two]]. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.<ref name="Altshiller-Court 1925 73">{{harvtxt|Altshiller-Court|1925|p=73}}</ref>

The center of the incircle, called the '''[[incenter]]''', can be found as the intersection of the three [[internal and external angle|internal]] [[angle bisector]]s.<ref name="Altshiller-Court 1925 73"/><ref>{{harvtxt|Kay|1969|p=117}}</ref> The center of an excircle is the intersection of the internal bisector of one angle (at vertex {{mvar|A}}, for example) and the [[internal and external angle|external]] bisectors of the other two. The center of this excircle is called the '''excenter''' relative to the vertex {{mvar|A}}, or the '''excenter''' of {{mvar|A}}.<ref name="Altshiller-Court 1925 73"/> Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an [[orthocentric system]].{{sfn|Johnson|1929|p=182}}