View source for Incircle and excircles

{{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}}

==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>

==Excircles and excenters==
[[File:Incircle and Excircles.svg|right|thumb|300px|
{{legend-line|solid black|[[Extended side]]s of {{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)}}
]]

An '''excircle''' or '''escribed circle'''<ref name="Altshiller-Court 1925 74"/> 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"/>

The center of an excircle is the intersection of the internal bisector of one angle (at vertex <math>A</math>, 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 <math>A</math>, or the '''excenter''' of <math>A</math>.<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}}

===Trilinear coordinates of excenters===
While the [[#incenter|incenter]] of <math>\triangle ABC</math> has [[trilinear coordinates]] <math>1 : 1 : 1</math>, the excenters have trilinears
{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{rrcrcr}
  J_A = & -1 &:& 1 &:& 1 \\
  J_B = & 1 &:& -1 &:& 1 \\
  J_C = & 1 &:& 1 &:& -1
\end{array}</math>

===Exradii===
The radii of the excircles are called the '''exradii'''.

The exradius of the excircle opposite <math>A</math> (so touching <math>BC</math>, centered at <math>J_A</math>) is<ref name="Altshiller-Court 1925 79">{{harvtxt|Altshiller-Court|1925|p=79}}</ref><ref>{{harvtxt|Kay|1969|p=202}}</ref>
:<math display=block>r_a = \frac{rs}{s - a} = \sqrt{\frac{s(s - b)(s - c)}{s - a}},</math> where <math>s = \tfrac{1}{2}(a + b + c).</math>

See [[Heron's formula]].

====Derivation of exradii formula====
Source:<ref name="Altshiller-Court 1925 79"/>

Let the excircle at side <math>AB</math> touch at side <math>AC</math> extended at <math>G</math>, and let this excircle's
radius be <math>r_c</math> and its center be <math>J_c</math>. Then <math>J_c G</math> is an altitude of <math>\triangle ACJ_c</math>, so <math>\triangle ACJ_c</math> has area <math>\tfrac12 br_c</math>. By a similar argument, <math>\triangle BCJ_c</math> has area <math>\tfrac12 ar_c</math> and <math>\triangle ABJ_c</math> has area <math>\tfrac12 cr_c</math>. Thus the area <math>\Delta</math> of triangle <math>\triangle ABC</math> is
:<math display=block>\Delta = \tfrac12 (a + b - c)r_c = (s - c)r_c</math>.

So, by symmetry, denoting <math>r</math> as the radius of the incircle,
:<math display=block>\Delta = sr = (s - a)r_a = (s - b)r_b = (s - c)r_c</math>.

By the [[Law of Cosines]], we have
:<math display=block>\cos A = \frac{b^2 + c^2 - a^2}{2bc}</math>

Combining this with the identity <math>\sin^2 \! A + \cos^2 \! A = 1</math>, we have
:<math display=block>\sin A = \frac{\sqrt{-a^4 - b^4 - c^4 + 2a^2 b^2 + 2b^2 c^2 + 2 a^2 c^2}}{2bc}</math>

But <math>\Delta = \tfrac12 bc \sin A</math>, and so
:<math display=block>\begin{align}
  \Delta &= \tfrac14 \sqrt{-a^4 - b^4 - c^4 + 2a^2b^2 + 2b^2 c^2 + 2 a^2 c^2} \\[5mu]
         &= \tfrac14 \sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)} \\[5mu]
         & = \sqrt{s(s - a)(s - b)(s - c)},
\end{align}</math>

which is [[Heron's formula]].

Combining this with <math>sr = \Delta</math>, we have
:<math display=block>r^2 = \frac{\Delta^2}{s^2} = \frac{(s - a)(s - b)(s - c)}{s}.</math>

Similarly, <math>(s - a)r_a = \Delta</math> gives
:<math display=block>\begin{align}
  &r_a^2 = \frac{s(s - b)(s - c)}{s - a} \\[4pt]
  &\implies r_a = \sqrt{\frac{s(s - b)(s - c)}{s - a}}.
\end{align}</math>

====Other properties====
From the formulas above one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas yields:<ref>Baker, Marcus, "A collection of formulae for the area of a plane triangle", ''Annals of Mathematics'', part 1 in vol. 1(6), January 1885, 134-138. (See also part 2 in vol. 2(1), September 1885, 11-18.)</ref>
:<math display=block>\Delta = \sqrt{r r_a r_b r_c}.</math>

===Other excircle properties===
The circular [[convex hull|hull]] of the excircles is internally tangent to each of the excircles and is thus an [[Problem of Apollonius|Apollonius circle]].<ref>[http://forumgeom.fau.edu/FG2002volume2/FG200222.pdf Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", ''Forum Geometricorum'' 2, 2002: pp. 175-182.]</ref> The radius of this Apollonius circle is <math>\tfrac{r^2 + s^2}{4r}</math> where <math>r</math> is the incircle radius and <math>s</math> is the semiperimeter of the triangle.<ref>[http://forumgeom.fau.edu/FG2003volume3/FG200320.pdf Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", ''Forum Geometricorum'' 3, 2003, 187-195.]</ref>

The following relations hold among the inradius <math>r</math>, the circumradius <math>R</math>, the semiperimeter <math>s</math>, and the excircle radii <math>r_a</math>, <math>r_b</math>, <math>r_c</math>:<ref name=Bell>{{cite web |author=Bell, Amy |title="Hansen's right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342. |url=http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |access-date=2012-05-05 |url-status=dead |archive-url=https://web.archive.org/web/20210831080348/https://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |archive-date=2021-08-31}}</ref>
:<math display=block>\begin{align}
              r_a + r_b + r_c &= 4R + r, \\
  r_a r_b + r_b r_c + r_c r_a &= s^2, \\
        r_a^2 + r_b^2 + r_c^2 &= \left(4R + r\right)^2 - 2s^2.
\end{align}</math>

The circle through the centers of the three excircles has radius <math>2R</math>.<ref name=Bell/>

If <math>H</math> is the [[orthocenter]] of <math>\triangle ABC</math>, then<ref name=Bell/>
:<math display=block>\begin{align}
          r_a + r_b + r_c + r &= \overline{AH} + \overline{BH} + \overline{CH} + 2R, \\
  r_a^2 + r_b^2 + r_c^2 + r^2 &= \overline{AH}^2 + \overline{BH}^2 + \overline{CH}^2 + (2R)^2.
\end{align}</math>

===Nagel triangle and Nagel point===
{{Main|Extouch triangle}}
[[File:Extouch Triangle and Nagel Point.svg|right|frame|
{{legend-line|solid black|[[Extended side]]s of triangle {{math|△''ABC''}}}}
{{legend-line|solid orange|Excircles of {{math|△''ABC''}} (tangent at {{mvar|T{{sub|A}}. T{{sub|B}}, T{{sub|C}}}})}}
{{legend-line|solid red|'''Nagel/Extouch triangle''' {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}}}}
{{legend-line|solid #728fce|[[Splitter (geometry)|Splitters]]: lines connecting opposite vertices of {{math|△''ABC''}} and {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}} (concur at '''Nagel point''' {{mvar|N}})}}
]]

The '''Nagel triangle''' or '''extouch triangle''' of <math>\triangle ABC</math> is denoted by the vertices <math>T_A</math>, <math>T_B</math>, and <math>T_C</math> that are the three points where the excircles touch the reference <math>\triangle ABC</math> and where <math>T_A</math> is opposite of <math>A</math>, etc. This <math>\triangle T_AT_BT_C</math> is also known as the '''extouch triangle''' of <math>\triangle ABC</math>. The [[circumcircle]] of the extouch <math>\triangle T_AT_BT_C</math> is called the '''Mandart circle'''
(cf. [[Mandart inellipse]]).

The three line segments <math>\overline{AT_A}</math>, <math>\overline{BT_B}</math> and <math>\overline{CT_C}</math> are called the [[splitter (geometry)|splitters]] of the triangle; they each bisect the perimeter of the triangle,{{Citation needed|date=May 2020}}
:<math display=block>\overline{AB} + \overline{BT_A} = \overline{AC} + \overline{CT_A} = \frac{1}{2}\left( \overline{AB} + \overline{BC} + \overline{AC} \right).</math>

The splitters intersect in a single point, the triangle's [[Nagel point]] <math>N_a</math> (or [[triangle center]] ''X''<sub>8</sub>).

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

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

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

The Nagel point is the [[isotomic conjugate]] of the Gergonne point.{{Citation needed|date=May 2020}}

==Related constructions==

===Nine-point circle and Feuerbach point===
{{Main|Nine-point circle}}

[[File:Circ9pnt3.svg|right|thumb|250px|The nine-point circle is tangent to the incircle and excircles]]

In [[geometry]], the '''nine-point circle''' is a [[circle]] that can be constructed for any given [[triangle]]. It is so named because it passes through nine significant [[concyclic points]] defined from the triangle. These nine [[point (geometry)|points]] are:<ref>{{harvtxt|Altshiller-Court|1925|pp=103–110}}</ref><ref>{{harvtxt|Kay|1969|pp=18,245}}</ref>

* The [[midpoint]] of each side of the triangle
* The [[perpendicular|foot]] of each [[altitude (triangle)|altitude]]
* The midpoint of the [[line segment]] from each [[vertex (geometry)|vertex]] of the triangle to the [[orthocenter]] (where the three altitudes meet; these line segments lie on their respective altitudes).

In 1822, Karl Feuerbach discovered that any triangle's nine-point circle is externally [[tangent circles|tangent]] to that triangle's three excircles and internally tangent to its incircle; this result is known as [[Feuerbach's theorem]]. He proved that:<ref>{{citation |ref={{harvid|Feuerbach|1822}} |last1=Feuerbach |first1=Karl Wilhelm |author1-link=Karl Wilhelm Feuerbach |last2=Buzengeiger |first2=Carl Heribert Ignatz |year=1822 |title=Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung |publisher=Wiessner |location=Nürnberg |edition=Monograph |language=de |url=https://gdz.sub.uni-goettingen.de/id/PPN512512426}}.</ref>
:... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle ... {{harv|Feuerbach|1822}}

The [[triangle center]] at which the incircle and the nine-point circle touch is called the [[Feuerbach point]].

===Incentral and excentral triangles===
The points of intersection of the interior angle bisectors of <math>\triangle ABC</math> with the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> are the vertices of the '''incentral triangle'''. Trilinear coordinates for the vertices of the incentral triangle <math>\triangle A'B'C'</math> are given by{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{ccccccc}
  A' &=& 0 &:& 1 &:& 1 \\[2pt]
  B' &=& 1 &:& 0 &:& 1 \\[2pt]
  C' &=& 1 &:& 1 &:& 0
\end{array}</math>

The '''excentral triangle''' of a reference triangle has vertices at the centers of the reference triangle's excircles. Its sides are on the external angle bisectors of the reference triangle (see figure at [[#top|top of page]]). Trilinear coordinates for the vertices of the excentral triangle <math>\triangle A'B'C'</math> are given by{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{ccrcrcr}
  A' &=& -1 &:& 1 &:& 1\\[2pt]
  B' &=& 1 &:& -1 &:& 1 \\[2pt]
  C' &=& 1 &:& 1 &:& -1
\end{array}</math>

==Equations for four circles==
Let <math>x:y:z</math> be a variable point in [[trilinear coordinates]], and let <math>u=\cos^2\left ( A/2 \right )</math>, <math>v=\cos^2\left ( B/2 \right )</math>, <math>w=\cos^2\left ( C/2 \right )</math>. The four circles described above are given equivalently by either of the two given equations:<ref name=WW>Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866).  https://www.forgottenbooks.com/en/search?q=%22Trilinear+coordinates%22</ref>{{rp|210–215}}
* Incircle:<math display=block>\begin{align}
  u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz - 2wuzx - 2uvxy &= 0 \\[4pt]
  {\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
* <math>A</math>-excircle:<math display=block>\begin{align}
  u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz + 2wuzx + 2uvxy &= 0 \\[4pt]
  {\textstyle \pm\sqrt{-x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
* <math>B</math>-excircle:<math display=block>\begin{align}
  u^2 x^2 + v^2 y^2 + w^2 z^2 + 2vwyz - 2wuzx + 2uvxy &= 0 \\[4pt]
  {\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{-y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>
* <math>C</math>-excircle:<math display=block>\begin{align}
  u^2 x^2 + v^2 y^2 + w^2 z^2 + 2vwyz + 2wuzx - 2uvxy &= 0 \\[4pt]
  {\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t}}\cos\tfrac{B}{2} \pm \sqrt{-z}\cos\tfrac{C}{2}} &= 0
\end{align}</math>

==Euler's theorem==
[[Euler's theorem in geometry|Euler's theorem]] states that in a triangle:
:<math display=block>(R - r)^2 = d^2 + r^2,</math>

where <math>R</math> and <math>r</math> are the circumradius and inradius respectively, and <math>d</math> is the distance between the [[circumcenter]] and the incenter.

For excircles the equation is similar:
:<math display=block>\left(R + r_\text{ex}\right)^2 = d_\text{ex}^2 + r_\text{ex}^2,</math>

where <math>r_\text{ex}</math> is the radius of one of the excircles, and <math>d_\text{ex}</math> is the distance between the circumcenter and that excircle's center.<ref name=Nelson>Nelson, Roger, "Euler's triangle inequality via proof without words", ''Mathematics Magazine'' 81(1), February 2008, 58-61.</ref>{{sfn|Johnson|1929|p=187}}<ref>[http://forumgeom.fau.edu/FG2001volume1/FG200120.pdf Emelyanov, Lev, and Emelyanova, Tatiana. "Euler's formula and Poncelet's porism", ''Forum Geometricorum'' 1, 2001: pp. 137–140.]</ref>

==Generalization to other polygons==
Some (but not all) [[quadrilateral]]s have an incircle. These are called [[tangential quadrilateral]]s. Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. This is called the [[Pitot theorem]].<ref>{{harvtxt|Josefsson|2011|loc=See in particular pp.&nbsp;65–66.}}</ref>

More generally, a polygon with any number of sides that has an inscribed circle (that is, one that is tangent to each side) is called a [[tangential polygon]].

==See also==
* {{annotated link|Circumcenter}}
* {{annotated link|Circumcircle}}
* {{annotated link|Circumconic and inconic}}
* {{annotated link|Circumgon}}
* {{annotated link|Ex-tangential quadrilateral}}
* {{annotated link|Harcourt's theorem}}
* {{annotated link|Incenter–excenter lemma}}
* {{annotated link|Inscribed sphere}}
* {{annotated link|Power of a point}}
* {{annotated link|Steiner inellipse}}
* {{annotated link|Tangential quadrilateral}}
* [[Triangle conic]]

==Notes==
{{Reflist|30em}}

==References==
* {{citation |last1=Altshiller-Court |first1=Nathan |year=1925 |lccn=52013504 |title=College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle |edition=2nd |publisher=[[Barnes & Noble]] |location=New York}}
* {{citation
|last=Johnson |first=Roger A.
|year=1929
|title=Modern Geometry
|publisher=Houghton Mifflin
|chapter=X. Inscribed and Escribed Circles
|pages=182–194
|chapter-url=https://archive.org/details/moderngeometry0000unse_q5z5/page/182/
|chapter-url-access=limited
}}
* {{citation |last=Josefsson|first=Martin|title=More characterizations of tangential quadrilaterals|journal=Forum Geometricorum|volume=11 |year=2011 |pages=65–82 |mr=2877281|url=http://forumgeom.fau.edu/FG2011volume11/FG201108.pdf |access-date=2023-03-14 |url-status=dead |archive-url=https://web.archive.org/web/20160304022959/http://forumgeom.fau.edu/FG2011volume11/FG201108.pdf |archive-date=2016-03-04}}
* {{citation |last1=Kay |first1=David C. |year=1969 |lccn=69012075 |title=College Geometry |publisher=[[Holt, Rinehart, and Winston]] |location=New York}}
* {{cite journal |last=Kimberling |first=Clark |title=Triangle Centers and Central Triangles |journal=Congressus Numerantium |issue=129 |year=1998 |pages=i-xxv,1–295}}
* {{cite journal |last=Kiss |first=Sándor |title=The Orthic-of-Intouch and Intouch-of-Orthic Triangles |journal=Forum Geometricorum |issue=6 |year=2006 |pages=171–177}}

==External links==
* [https://mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-incircle Derivation of formula for radius of incircle of a triangle], MATHalino
* {{MathWorld|title=Incircle|urlname=Incircle}}

===Interactive===
* [https://www.mathopenref.com/triangleincenter.html Triangle incenter]; [https://www.mathopenref.com/triangleincircle.html Triangle incircle]; [https://www.mathopenref.com/polygonincircle.html Incircle of a regular polygon] (with interactive animations)
* [https://www.mathopenref.com/constincircle.html Constructing a triangle's incenter / incircle with compass and straightedge] An interactive animated demonstration
* [https://www.cut-the-knot.org/Curriculum/Geometry/AdjacentIncircles.shtml Equal Incircles Theorem] at [[cut-the-knot]]
* [https://www.cut-the-knot.org/Curriculum/Geometry/FourIncircles.shtml Five Incircles Theorem] at [[cut-the-knot]]
* [https://www.cut-the-knot.org/Curriculum/Geometry/IncirclesInQuadri.shtml Pairs of Incircles in a Quadrilateral] at [[cut-the-knot]]
* [https://web.archive.org/web/20151105214641/http://www.uff.br/trianglecenters/X0001.html An interactive Java applet for the incenter]

[[Category:Circles defined for a triangle]]