Editing Incircle and excircles (section)

==Incircle and Incenter==
{{See also|Incenter}}

Suppose <math>\triangle ABC</math> has an incircle with radius <math>r</math> and center <math>I</math>. Let <math>a</math> be the length of <math>\overline{BC}</math>, <math>b</math> the length of <math>\overline{AC}</math>, and <math>c</math> the length of <math>\overline{AB}</math>.

Also let <math>T_A</math>, <math>T_B</math>, and <math>T_C</math> be the touchpoints where the incircle touches <math>\overline{BC}</math>, <math>\overline{AC}</math>, and <math>\overline{AB}</math>.

===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"/>

===Incircle and its radius properties===

====Distances between vertex and nearest touchpoints====
The distances from a vertex to the two nearest touchpoints are equal; for example:<ref name=":0">''Mathematical Gazette'', July 2003, 323-324.</ref>
:<math display=block>d\left(A, T_B\right) = d\left(A, T_C\right) = \tfrac12(b + c - a) = s - a.</math>

====Other properties====
If the [[altitude (triangle)|altitudes]] from sides of lengths <math>a</math>, <math>b</math>, and <math>c</math> are <math>h_a</math>, <math>h_b</math>, and <math>h_c</math>, then the inradius <math>r</math> is one-third of the [[harmonic mean]] of these altitudes; that is,<ref>{{harvtxt|Kay|1969|p=203}}</ref>
:<math display=block> r = \frac{1}{\dfrac{1}{h_a} + \dfrac{1}{h_b} + \dfrac{1}{h_c}}.</math>

The product of the incircle radius <math>r</math> and the [[circumcircle]] radius <math>R</math> of a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> is{{sfn|Johnson|1929|p=189, #298(d)}}
:<math display=block>rR = \frac{abc}{2(a + b + c)}.</math>

Some relations among the sides, incircle radius, and circumcircle radius are:<ref name=Bell/>
:<math display=block>\begin{align}
     ab + bc + ca &=  s^2 +  (4R + r)r, \\
  a^2 + b^2 + c^2 &= 2s^2 - 2(4R + r)r.
\end{align}</math>

Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.<ref>Kodokostas, Dimitrios, "Triangle Equalizers", ''Mathematics Magazine'' 83, April 2010, pp. 141-146.</ref>

The incircle radius is no greater than one-ninth the sum of the altitudes.<ref>Posamentier, Alfred S., and Lehmann, Ingmar. ''[[The Secrets of Triangles]]'', Prometheus Books, 2012.</ref>{{rp|289}}

The squared distance from the incenter <math>I</math> to the [[circumcenter]] <math>O</math> is given by<ref name=Franzsen>{{cite journal
|last=Franzsen
|first=William N.
|journal=Forum Geometricorum
|mr=2877263
|pages=231–236
|title=The distance from the incenter to the Euler line
|volume=11
|year=2011
|url=http://forumgeom.fau.edu/FG2011volume11/FG201126.pdf
|access-date=2012-05-09
|url-status=dead
|archive-url=https://web.archive.org/web/20201205220605/http://forumgeom.fau.edu/FG2011volume11/FG201126.pdf
|archive-date=2020-12-05
}}.</ref>{{rp|232}}
:<math display=block>\overline{OI}^2 = R(R - 2r) = \frac{a\,b\,c\,}{a+b+c}\left [\frac{a\,b\,c\,}{(a+b-c)\,(a-b+c)\,(-a+b+c)}-1 \right ]</math>

and the distance from the incenter to the center <math>N</math> of the [[nine point circle]] is<ref name=Franzsen/>{{rp|232}}
:<math display=block>\overline{IN} = \tfrac12(R - 2r) < \tfrac12 R.</math>

The incenter lies in the [[medial triangle]] (whose vertices are the midpoints of the sides).<ref name=Franzsen/>{{rp|233, Lemma 1}}

====Relation to area of the triangle====
{{Redirect|Inradius|the three-dimensional equivalent|Inscribed sphere}}
The radius of the incircle is related to the [[area]] of the triangle.<ref>Coxeter, H.S.M. "Introduction to Geometry'' 2nd ed. Wiley, 1961.''</ref> The ratio of the area of the incircle to the area of the triangle is less than or equal to <math>\pi \big/ 3\sqrt3</math>,
with equality holding only for [[equilateral triangle]]s.<ref>Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", ''[[American Mathematical Monthly]]'' 115, October 2008, 679-689: Theorem 4.1.</ref>

Suppose <math>\triangle ABC</math> has an incircle with radius <math>r</math> and center <math>I</math>. Let <math>a</math> be the length of <math>\overline{BC}</math>, <math>b</math> the length of <math>\overline{AC}</math>, and <math>c</math> the length of <math>\overline{AB}</math>.

Now, the incircle is tangent to <math>\overline{AB}</math> at some point <math>T_C</math>, and so <math>\angle AT_CI</math> is right. Thus, the radius <math>T_CI</math> is an [[altitude (triangle)|altitude]] of <math>\triangle IAB</math>.

Therefore, <math>\triangle IAB</math> has base length <math>c</math> and height <math>r</math>, and so has area <math>\tfrac12 cr</math>.

Similarly, <math>\triangle IAC</math> has area <math>\tfrac12 br</math> and <math>\triangle IBC</math> has area <math>\tfrac12 ar</math>.

Since these three triangles decompose <math>\triangle ABC</math>, we see that the area <math>\Delta \text{ of} \triangle ABC</math> is:
:<math display=block>\Delta = \tfrac12 (a + b + c)r = sr,</math>

{{spaces|4}} and {{spaces|4}}<math>r = \frac{\Delta}{s},</math>

where <math>\Delta</math> is the area of <math>\triangle ABC</math> and <math>s = \tfrac12(a + b + c)</math> is its [[semiperimeter]].

For an alternative formula, consider <math>\triangle IT_CA</math>. This is a right-angled triangle with one side equal to <math>r</math> and the other side equal to <math>r \cot \tfrac{A}{2}</math>. The same is true for <math>\triangle IB'A</math>. The large triangle is composed of six such triangles and the total area is:{{Citation needed|date=May 2020}}
:<math display=block>\Delta = r^2 \left(\cot\tfrac{A}{2} + \cot\tfrac{B}{2} + \cot\tfrac{C}{2}\right).</math>

===Gergonne triangle and point===
[[File:Intouch Triangle and Gergonne Point.svg|right|frame|
{{legend-line|solid black|Triangle {{math|△''ABC''}}}}
{{legend-line|solid #728fce|Incircle ([[incenter]] at {{mvar|I}})}}
{{legend-line|solid red|Contact triangle {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}}}}
{{legend-line|solid #1dc404|Lines between opposite vertices of {{math|△''ABC''}} and {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}} (concur at Gergonne point {{mvar|G{{sub|e}}}})}}
]]

The '''Gergonne triangle''' (of <math>\triangle ABC</math>) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite <math>A</math> is denoted <math>T_A</math>, etc.

This Gergonne triangle, <math>\triangle T_AT_BT_C</math>, is also known as the '''contact triangle''' or '''intouch triangle''' of <math>\triangle ABC</math>. Its area is
:<math display=block>K_T = K\frac{2r^2 s}{abc}</math>

where <math>K</math>, <math>r</math>, and <math>s</math> are the area, radius of the incircle, and semiperimeter of the original triangle, and <math>a</math>, <math>b</math>, and <math>c</math> are the side lengths of the original triangle. This is the same area as that of the [[extouch triangle]].<ref>
Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ContactTriangle.html</ref>

The three lines <math>AT_A</math>, <math>BT_B</math>, and <math>CT_C</math> intersect in a single point called the '''Gergonne point''', denoted as <math>G_e</math> (or [[triangle center]] ''X''<sub>7</sub>). The Gergonne point lies in the open [[orthocentroidal disk]] punctured at its own center, and can be any point therein.<ref name=Bradley>Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", ''[[Forum Geometricorum]]'' 6 (2006), 57–70. http://forumgeom.fau.edu/FG2006volume6/FG200607index.html</ref>

The Gergonne point of a triangle has a number of properties, including that it is the [[symmedian point]] of the Gergonne triangle.<ref>
{{cite journal
|last=Dekov
|first=Deko
|title=Computer-generated Mathematics : The Gergonne Point
|journal=Journal of Computer-generated Euclidean Geometry
|year=2009
|volume=1
|pages=1&ndash;14
|url=http://www.dekovsoft.com/j/2009/01/JCGEG200901.pdf
|url-status=dead
|archive-url=https://web.archive.org/web/20101105045604/http://www.dekovsoft.com/j/2009/01/JCGEG200901.pdf
|archive-date=2010-11-05
}}</ref>

[[Trilinear coordinates]] for the vertices of the intouch triangle are given by{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{ccccccc}
  T_A &=& 0 &:& \sec^2 \frac{B}{2} &:& \sec^2\frac{C}{2} \\[2pt]
  T_B &=& \sec^2 \frac{A}{2} &:& 0 &:& \sec^2\frac{C}{2} \\[2pt]
  T_C &=& \sec^2 \frac{A}{2} &:& \sec^2\frac{B}{2} &:& 0.
\end{array}</math>

Trilinear coordinates for the Gergonne point are given by{{Citation needed|date=May 2020}}
:<math display=block>\sec^2\tfrac{A}{2} : \sec^2\tfrac{B}{2} : \sec^2\tfrac{C}{2},</math>

or, equivalently, by the [[Law of Sines]],
:<math display=block>\frac{bc}{b + c - a} : \frac{ca}{c + a - b} : \frac{ab}{a + b - c}.</math>