Editing Incircle and excircles (section)

===Incenter===
The incenter is the point where the internal [[angle bisectors]] of <math>\angle ABC, \angle BCA, \text{ and } \angle BAC</math> meet.

====Trilinear coordinates====
The [[trilinear coordinates]] for a point in the triangle is the ratio of all the distances to the triangle sides. Because the incenter is the same distance from all sides of the triangle, the trilinear coordinates for the incenter are<ref name="etc">[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers] {{webarchive|url=https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |date=2012-04-19}}, accessed 2014-10-28.</ref>
:<math display=block>\ 1 : 1 : 1.</math>

====Barycentric coordinates====
The [[barycentric coordinates (mathematics)|barycentric coordinates]] for a point in a triangle give weights such that the point is the weighted average of the triangle vertex positions.
Barycentric coordinates for the incenter are given by
:<math display=block>a : b : c</math>

where <math>a</math>, <math>b</math>, and <math>c</math> are the lengths of the sides of the triangle, or equivalently (using the [[law of sines]]) by
:<math display=block>\sin A : \sin B : \sin C</math>

where <math>A</math>, <math>B</math>, and <math>C</math> are the angles at the three vertices.

====Cartesian coordinates====
The [[Cartesian coordinates]] of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter (that is, using the barycentric coordinates given above, normalized to sum to unity) as weights. The weights are positive so the incenter lies inside the triangle as stated above. If the three vertices are located at <math>(x_a,y_a)</math>, <math>(x_b,y_b)</math>, and <math>(x_c,y_c)</math>, and the sides opposite these vertices have corresponding lengths <math>a</math>, <math>b</math>, and <math>c</math>, then the incenter is at{{Citation needed|date=May 2020}}
:<math display=block>
    \left(\frac{a x_a + b x_b + c x_c}{a + b + c}, \frac{a y_a + b y_b + c y_c}{a + b + c}\right)
  = \frac{a\left(x_a, y_a\right) + b\left(x_b, y_b\right) + c\left(x_c, y_c\right)}{a + b + c}.
</math>

====Radius====
The inradius <math>r</math> of the incircle in a triangle with sides of length <math>a</math>, <math>b</math>, <math>c</math> is given by<ref>{{harvtxt|Kay|1969|p=201}}</ref>
:<math display=block>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}},</math>

where <math>s = \tfrac12(a + b + c)</math> is the semiperimeter (see [[Heron's formula]]).

The tangency points of the incircle divide the sides into segments of lengths <math>s-a</math> from <math>A</math>, <math>s-b</math> from <math>B</math>, and <math>s-c</math> from <math>C</math> (see [[Tangent_lines_to_circles#Tangent_lines_to_one_circle|Tangent lines to a circle]]).<ref>Chu, Thomas, ''The Pentagon'', Spring 2005, p. 45, problem 584.</ref>

====Distances to the vertices====
Denote the incenter of <math>\triangle ABC</math> as <math>I</math>.

The distance from vertex <math>A</math> to the incenter <math>I</math> is:
:<math display=block>
    \overline{AI} = d(A, I)
  = c \, \frac{\sin\frac{B}{2}}{\cos\frac{C}{2}}
  = b \, \frac{\sin\frac{C}{2}}{\cos\frac{B}{2}}.
</math>

====Derivation of the formula stated above====
Use the [[Law of sines]] in the triangle <math>\triangle IAB</math>.

We get <math>\frac{\overline{AI}}{\sin \frac{B}{2}} = \frac{c}{\sin \angle AIB}</math>.
We have that <math>\angle AIB = \pi - \frac{A}{2} - \frac{B}{2} = \frac{\pi}{2} + \frac{C}{2}</math>.

It follows that <math>\overline{AI} = c \ \frac{\sin \frac{B}{2}}{\cos \frac{C}{2}}</math>.

The equality with the second expression is obtained the same way.

The distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation<ref>
{{citation
|last1=Allaire |first1=Patricia R.
|last2=Zhou |first2=Junmin
|last3=Yao |first3=Haishen
|date=March 2012
|journal=[[Mathematical Gazette]]
|pages=161–165
|title=Proving a nineteenth century ellipse identity
|volume=96
|doi=10.1017/S0025557200004277
|s2cid=124176398
}}.</ref>
:<math display=block>\frac{\overline{IA} \cdot \overline{IA}}{\overline{CA} \cdot \overline{AB}} + \frac{\overline{IB} \cdot \overline{IB}}{\overline{AB} \cdot \overline{BC}} + \frac{\overline{IC} \cdot \overline{IC}}{\overline{BC} \cdot \overline{CA}} = 1.</math>

Additionally,<ref>{{citation |last=Altshiller-Court |first=Nathan |author-link=Nathan Altshiller Court |title=College Geometry |publisher=Dover Publications |year=1980}}. #84, p.&nbsp;121.</ref>
:<math display=block>\overline{IA} \cdot \overline{IB} \cdot \overline{IC} = 4Rr^2,</math>

where <math>R</math> and <math>r</math> are the triangle's [[circumradius]] and [[inradius]] respectively.

====Other properties====
The collection of triangle centers may be given the structure of a [[group (mathematics)|group]] under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the [[identity element]].<ref name="etc"/>