Editing Incenter (section)

==Relation to triangle sides and vertices==

===Trilinear coordinates===
The [[trilinear coordinates]] for a point in the triangle give the ratio of distances to the triangle sides. Trilinear coordinates
for the incenter are given by<ref name="etc"/>
:<math>1 : 1 : 1.</math>

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

===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>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>\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—i.e., 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
:<math>\bigg(\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}\bigg) = \frac{a(x_A,y_A)+b(x_B,y_B)+c(x_C,y_C)}{a+b+c}.</math>

===Distances to vertices===
Denoting the incenter of triangle ''ABC'' as ''I'', 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| issue = 535
 | doi = 10.1017/S0025557200004277
 }}.</ref>
:<math>\frac{IA\cdot IA}{CA \cdot AB}+ \frac{IB \cdot IB}{AB\cdot BC} + \frac{IC \cdot IC}{BC\cdot CA} = 1.</math>
Additionally,<ref name=ac>{{citation|last=Altshiller-Court|first=Nathan|authorlink=Nathan Altshiller Court|title=College Geometry|publisher=Dover Publications|year=1980}}. #84, p.&nbsp;121.</ref>
:<math>IA \cdot IB \cdot IC=4Rr^2,</math>
where ''R'' and ''r'' are the triangle's [[circumradius]] and [[inradius]] respectively.