($I$), excircles, excenters ($J\_A$, $J\_B$, $J\_C$), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle.
In geometry, the incircle or inscribed circle of a triangle is the largest circle 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.
An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex $A$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex $A$, or the excenter of $A$. 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.
All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons. See also Tangent lines to circles.

Incircle and incenter

Suppose $\backslash triangle\; ABC$ has an incircle with radius $r$ and center $I$. Let $a$ be the length of $BC$, $b$ the length of $AC$, and $c$ the length of $AB$. Also let $T\_A$, $T\_B$, and $T\_C$ be the touchpoints where the incircle touches $BC$, $AC$, and $AB$.

Incenter

The incenter is the point where the internal angle bisectors of $\backslash angle\; ABC,\; \backslash angle\; BCA,\; \backslash text\; \backslash angle\; BAC$ meet. The distance from vertex $A$ to the incenter $I$ is: :$d(A,\; I)\; =\; c\; \backslash frac\; =\; b\; \backslash frac.$

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 areEncyclopedia of Triangle Centers

, accessed 2014-10-28. :$\backslash \; 1\; :\; 1\; :\; 1.$

Barycentric coordinates

The 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 :$\backslash \; a\; :\; b\; :\; c$ where $a$, $b$, and $c$ are the lengths of the sides of the triangle, or equivalently (using the law of sines) by :$\backslash sin(A):\backslash sin(B):\backslash sin(C)$ where $A$, $B$, and $C$ 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 $(x\_a,y\_a)$, $(x\_b,y\_b)$, and $(x\_c,y\_c)$, and the sides opposite these vertices have corresponding lengths $a$, $b$, and $c$, then the incenter is at :$\backslash left(\backslash frac,\; \backslash frac\backslash right)\; =\; \backslash frac.$

Radius

The inradius $r$ of the incircle in a triangle with sides of length $a$'', $b$'', $c$ is given by :$r\; =\; \backslash sqrt,$ where $s\; =\; (a\; +\; b\; +\; c)/2.$ See Heron's formula.

Distances to the vertices

Denoting the incenter of $\backslash triangle\; ABC$ as $I$, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation :$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 1.$ Additionally, :$IA\; \backslash cdot\; IB\; \backslash cdot\; IC\; =\; 4Rr^2,$ where $R$ and $r$ are the triangle's circumradius and inradius respectively.

Other properties

The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element.

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:''Mathematical Gazette'', July 2003, 323-324. :$d\backslash left(A,\; T\_B\backslash right)\; =\; d\backslash left(A,\; T\_C\backslash right)\; =\; \backslash frac(b\; +\; c\; -\; a).$

Other properties

Suppose the tangency points of the incircle divide the sides into lengths of $x$ and $y$, $y$ and $z$, and ''$z$'' and $x$. Then the incircle has the radius :$r\; =\; \backslash sqrt$ and the area of the triangle is :$\backslash Delta\; =\; \backslash sqrt.$ If the altitudes from sides of lengths $a$'', $b$'', and $c$ are $h\_a$, $h\_b$, and ''$h\_c$'', then the inradius ''$r$'' is one-third of the harmonic mean of these altitudes; that is, :$r\; =\; \backslash frac.$ The product of the incircle radius ''$r$'' and the circumcircle radius $R$ of a triangle with sides $a$'', $b$'', and $c$ isJohnson, Roger A., ''Advanced Euclidean Geometry'', Dover, 2007 (orig. 1929). :$rR\; =\; \backslash frac.$ Some relations among the sides, incircle radius, and circumcircle radius are: :$\backslash begin\; ab\; +\; bc\; +\; ca\; \&=\; s^2\; +\; (4R\; +\; r)r,\; \backslash \backslash \; a^2\; +\; b^2\; +\; c^2\; \&=\; 2s^2\; -\; 2(4R\; +\; r)r.\; \backslash end$ 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. Denoting the center of the incircle of $\backslash triangle\; ABC$ as $I$, we have :$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 1$ and :$IA\; \backslash cdot\; IB\; \backslash cdot\; IC\; =\; 4Rr^2.$ The incircle radius is no greater than one-ninth the sum of the altitudes. The squared distance from the incenter $I$ to the circumcenter $O$ is given by. :$OI^2\; =\; R(R\; -\; 2r)$, and the distance from the incenter to the center $N$ of the nine point circle is : $IN\; =\; \backslash frac(R\; -\; 2r)\; <\; \backslash fracR.$ The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).

Relation to area of the triangle

The radius of the incircle is related to the area of the triangle. The ratio of the area of the incircle to the area of the triangle is less than or equal to $\backslash tfrac$, with equality holding only for equilateral triangles. Suppose $\backslash triangle\; ABC$ has an incircle with radius $r$ and center $I$. Let $a$ be the length of $BC$, $b$ the length of $AC$, and $c$ the length of $AB$''.'' Now, the incircle is tangent to ''$AB$'' at some point $T\_C$, and so $\backslash angle\; AT\_CI$ is right. Thus, the radius $T\_CI$ is an altitude of $\backslash triangle\; IAB$. Therefore, $\backslash triangle\; IAB$ has base length ''$c$'' and height $r$, and so has area $\backslash tfraccr$. Similarly, $\backslash triangle\; IAC$ has area $\backslash tfracbr$ and $\backslash triangle\; IBC$ has area $\backslash tfracar$. Since these three triangles decompose $\backslash triangle\; ABC$, we see that the area $\backslash Delta\; \backslash text\; \backslash triangle\; ABC$ is: :$\backslash Delta\; =\; \backslash frac\; (a\; +\; b\; +\; c)r\; =\; sr,$ and $r\; =\; \backslash frac,$ where $\backslash Delta$ is the area of $\backslash triangle\; ABC$ and $s\; =\; \backslash tfrac(a\; +\; b\; +\; c)$ is its semiperimeter. For an alternative formula, consider $\backslash triangle\; IT\_CA$. This is a right-angled triangle with one side equal to ''$r$'' and the other side equal to $r\; \backslash cot\backslash left(\backslash frac\backslash right)$. The same is true for $\backslash triangle\; IB\text{'}A$. The large triangle is composed of six such triangles and the total area is: :$\backslash Delta\; =\; r^2\; \backslash left(\backslash cot\backslash left(\backslash frac\backslash right)\; +\; \backslash cot\backslash left(\backslash frac\backslash right)\; +\; \backslash cot\backslash left(\backslash frac\backslash right)\backslash right).$

Gergonne triangle and point

The Gergonne triangle (of ''$\backslash triangle\; ABC$'') is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite $A$ is denoted ''$T\_A$'', etc. This Gergonne triangle, ''$\backslash triangle\; T\_AT\_BT\_C$'', is also known as the contact triangle or intouch triangle of ''$\backslash triangle\; ABC$''. Its area is :$K\_T\; =\; K\backslash frac$ where $K$, $r$, and $s$ are the area, radius of the incircle, and semiperimeter of the original triangle, and $a$, $b$, and $c$ are the side lengths of the original triangle. This is the same area as that of the extouch triangle. The three lines ''$AT\_A$'', ''$BT\_B$'' and ''$CT\_C$'' intersect in a single point called the Gergonne point, denoted as ''$G\_e$'' (or triangle center ''X''_{7}). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.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
The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.
Trilinear coordinates for the vertices of the intouch triangle are given by
*$\backslash text\backslash ,\; T\_A\; =\; 0\; :\; \backslash sec^2\; \backslash left(\backslash frac\backslash right)\; :\; \backslash sec^2\backslash left(\backslash frac\backslash right)$
*$\backslash text\backslash ,\; T\_B\; =\; \backslash sec^2\; \backslash left(\backslash frac\backslash right)\; :\; 0\; :\; \backslash sec^2\backslash left(\backslash frac\backslash right)$
*$\backslash text\backslash ,\; T\_C\; =\; \backslash sec^2\; \backslash left(\backslash frac\backslash right)\; :\; \backslash sec^2\backslash left(\backslash frac\backslash right)\; :\; 0.$
Trilinear coordinates for the Gergonne point are given by
:$\backslash sec^2\backslash left(\backslash frac\backslash right)\; :\; \backslash sec^2\; \backslash left(\backslash frac\backslash right)\; :\; \backslash sec^2\backslash left(\backslash frac\backslash right),$
or, equivalently, by the Law of Sines,
:$\backslash frac\; :\; \backslash frac\; :\; \backslash frac.$

Excircles and excenters

$I$), excircles, excenters ($J\_A$, $J\_B$, $J\_C$), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle. An excircle or escribed circle 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. The center of an excircle is the intersection of the internal bisector of one angle (at vertex $A$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex $A$, or the excenter of $A$. 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.

Trilinear coordinates of excenters

While the incenter of ''$\backslash triangle\; ABC$'' has trilinear coordinates $1\; :\; 1\; :\; 1$, the excenters have trilinears $-1\; :\; 1\; :\; 1$, $1\; :\; -1\; :\; 1$, and $1\; :\; 1\; :\; -1$.

Exradii

The radii of the excircles are called the exradii. The exradius of the excircle opposite $A$ (so touching $BC$, centered at $J\_A$) is : $r\_a\; =\; \backslash frac\; =\; \backslash sqrt,$ where $s\; =\; \backslash tfrac(a\; +\; b\; +\; c).$ See Heron's formula.

Derivation of exradii formula

Let the excircle at side $AB$ touch at side $AC$ extended at $G$, and let this excircle's radius be $r\_c$ and its center be $J\_c$. Then $J\_c\; G$ is an altitude of $\backslash triangle\; ACJ\_c$, so $\backslash triangle\; ACJ\_c$ has area $\backslash tfracbr\_c$. By a similar argument, $\backslash triangle\; BCJ\_c$ has area $\backslash tfracar\_c$ and $\backslash triangle\; ABJ\_c$ has area $\backslash tfraccr\_c$. Thus the area $\backslash Delta$ of triangle $\backslash triangle\; ABC$ is : $\backslash Delta\; =\; \backslash frac(a\; +\; b\; -\; c)r\_c\; =\; (s\; -\; c)r\_c$. So, by symmetry, denoting $r$ as the radius of the incircle, : $\backslash Delta\; =\; sr\; =\; (s\; -\; a)r\_a\; =\; (s\; -\; b)r\_b\; =\; (s\; -\; c)r\_c$. By the Law of Cosines, we have : $\backslash cos(A)\; =\; \backslash frac$ Combining this with the identity $\backslash sin^2\; A\; +\; \backslash cos^2\; A\; =\; 1$, we have : $\backslash sin(A)\; =\; \backslash frac$ But $\backslash Delta\; =\; \backslash tfracbc\; \backslash sin(A)$, and so :$\backslash begin\; \backslash Delta\; \&=\; \backslash frac\; \backslash sqrt\; \backslash \backslash \; \&=\; \backslash frac\; \backslash sqrt\; \backslash \backslash \; \&\; =\; \backslash sqrt,\; \backslash end$ which is Heron's formula. Combining this with $sr\; =\; \backslash Delta$, we have :$r^2\; =\; \backslash frac\; =\; \backslash frac.$ Similarly, $(s\; -\; a)r\_a\; =\; \backslash Delta$ gives :$r\_a^2\; =\; \backslash frac$ and :$r\_a\; =\; \backslash sqrt.$

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: :$\backslash Delta\; =\; \backslash sqrt.$

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle. The radius of this Apollonius circle is $\backslash tfrac$ where $r$ is the incircle radius and $s$ is the semiperimeter of the triangle. The following relations hold among the inradius ''$r$'', the circumradius $R$, the semiperimeter ''$s$'', and the excircle radii ''$r\_a$'', ''$r\_b$'', ''$r\_c$'':Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342.

/ref> :$\backslash begin\; r\_a\; +\; r\_b\; +\; r\_c\; \&=\; 4R\; +\; r,\; \backslash \backslash \; r\_a\; r\_b\; +\; r\_b\; r\_c\; +\; r\_c\; r\_a\; \&=\; s^2,\; \backslash \backslash \; r\_a^2\; +\; r\_b^2\; +\; r\_c^2\; \&=\; \backslash left(4R\; +\; r\backslash right)^2\; -\; 2s^2.\; \backslash end$ The circle through the centers of the three excircles has radius $2R$. If ''$H$'' is the orthocenter of ''$\backslash triangle\; ABC$'', then :$\backslash begin\; r\_a\; +\; r\_b\; +\; r\_c\; +\; r\; \&=\; AH\; +\; BH\; +\; CH\; +\; 2R,\; \backslash \backslash \; r\_a^2\; +\; r\_b^2\; +\; r\_c^2\; +\; r^2\; \&=\; AH^2\; +\; BH^2\; +\; CH^2\; +\; (2R)^2.\; \backslash end$

** Nagel triangle and Nagel point **

frame|325px|The extouch triangle (''$\backslash triangle\; T\_AT\_BT\_C$'') and the Nagel point ($N$) of a triangle (''$\backslash triangle\; ABC$''). The orange circles are the [[excircles of the triangle.
The Nagel triangle or extouch triangle of ''$\backslash triangle\; ABC$'' is denoted by the vertices $T\_A$, $T\_B$, and $T\_C$ that are the three points where the excircles touch the reference ''$\backslash triangle\; ABC$'' and where ''$T\_A$'' is opposite of ''$A$'', etc. This ''$\backslash triangle\; T\_AT\_BT\_C$'' is also known as the extouch triangle of ''$\backslash triangle\; ABC$''. The circumcircle of the extouch ''$\backslash triangle\; T\_AT\_BT\_C$'' is called the Mandart circle.
The three lines $AT\_A$, $BT\_B$ and $CT\_C$ are called the splitters of the triangle; they each bisect the perimeter of the triangle,
:$AB\; +\; BT\_A\; =\; AC\; +\; CT\_A\; =\; \backslash frac\backslash left(\; AB\; +\; BC\; +\; AC\; \backslash right).$
The splitters intersect in a single point, the triangle's Nagel point $N\_a$ (or triangle center ''X''_{8}).
Trilinear coordinates for the vertices of the extouch triangle are given by
* $\backslash text\; \backslash ,\; T\_A\; =\; 0\; :\; \backslash csc^2\backslash left(\backslash frac\backslash right)\; :\; \backslash csc^2\backslash left(\backslash frac\backslash right)$
* $\backslash text\; \backslash ,\; T\_B\; =\; \backslash csc^2\backslash left(\backslash frac\backslash right)\; :\; 0\; :\; \backslash csc^2\backslash left(\backslash frac\backslash right)$
*$\backslash text\; \backslash ,\; T\_C\; =\; \backslash csc^2\backslash left(\backslash frac\backslash right)\; :\; \backslash csc^2\backslash left(\backslash frac\backslash right)\; :\; 0.$
Trilinear coordinates for the Nagel point are given by
:$\backslash csc^2\backslash left(\backslash frac\backslash right)\; :\; \backslash csc^2\; \backslash left(\backslash frac\backslash right)\; :\; \backslash csc^2\backslash left(\backslash frac\backslash right),$
or, equivalently, by the Law of Sines,
:$\backslash frac\; :\; \backslash frac\; :\; \backslash frac.$
The Nagel point is the isotomic conjugate of the Gergonne point.

Related constructions

Nine-point circle and Feuerbach point

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 points are: * The midpoint of each side of the triangle * The foot of each altitude * The midpoint of the line segment from each 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 to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that: :... 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 ... 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 ''$\backslash triangle\; ABC$'' with the segments ''$BC$, $CA$,'' and ''$AB$'' are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle are given by * $\backslash \; \backslash left(\; \backslash text\; \backslash ,\; A\backslash right)\; =\; 0\; :\; 1\; :\; 1$ * $\backslash \; \backslash left(\; \backslash text\; \backslash ,\; B\backslash right)\; =\; 1\; :\; 0\; :\; 1$ *$\backslash \; \backslash left(\; \backslash text\; \backslash ,\; C\backslash right)\; =\; 1\; :\; 1\; :\; 0.$ 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 of page). Trilinear coordinates for the vertices of the excentral triangle are given by * $(\backslash text\; \backslash ,\; A)\; =\; -1\; :\; 1\; :\; 1$ * $(\backslash text\; \backslash ,\; B)\; =\; 1\; :\; -1\; :\; 1$ *$(\backslash text\; \backslash ,\; C)\; =\; 1\; :\; 1\; :\; -1.$

Equations for four circles

Let ''$x:y:z$'' be a variable point in trilinear coordinates, and let ''$u=\backslash cos^2\backslash left\; (\; A/2\; \backslash right\; )$'', ''$v=\backslash cos^2\backslash left\; (\; B/2\; \backslash right\; )$'', ''$w=\backslash cos^2\backslash left\; (\; C/2\; \backslash right\; )$''. The four circles described above are given equivalently by either of the two given equations:Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books * Incircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; -\; 2vwyz\; -\; 2wuzx\; -\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$ *''$A$-''excircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; -\; 2vwyz\; +\; 2wuzx\; +\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$ * ''$B$-''excircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; +\; 2vwyz\; -\; 2wuzx\; +\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$ *''$C$-''excircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; +\; 2vwyz\; +\; 2wuzx\; -\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$

Euler's theorem

Euler's theorem states that in a triangle: :$(R\; -\; r)^2\; =\; d^2\; +\; r^2,$ where ''$R$'' and ''$r$'' are the circumradius and inradius respectively, and ''$d$'' is the distance between the circumcenter and the incenter. For excircles the equation is similar: :$\backslash left(R\; +\; r\_\backslash text\backslash right)^2\; =\; d\_\backslash text^2\; +\; r\_\backslash text^2,$ where ''$r\_\backslash text$'' is the radius of one of the excircles, and ''$d\_\backslash text$'' is the distance between the circumcenter and that excircle's center.Nelson, Roger, "Euler's triangle inequality via proof without words," ''Mathematics Magazine'' 81(1), February 2008, 58-61.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) quadrilaterals have an incircle. These are called tangential quadrilaterals. 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. 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

* * * * * * * * * *

Notes

References

* * * *

External links

Derivation of formula for radius of incircle of a triangle

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Interactive

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nbsp; With interactive animations

An interactive animated demonstration

Equal Incircles Theorem

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Five Incircles Theorem

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Pairs of Incircles in a Quadrilateral

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An interactive Java applet for the incenter

{{DEFAULTSORT:Incircle And Excircles Of A Triangle Category:Circles defined for a triangle

Incircle and incenter

Suppose $\backslash triangle\; ABC$ has an incircle with radius $r$ and center $I$. Let $a$ be the length of $BC$, $b$ the length of $AC$, and $c$ the length of $AB$. Also let $T\_A$, $T\_B$, and $T\_C$ be the touchpoints where the incircle touches $BC$, $AC$, and $AB$.

Incenter

The incenter is the point where the internal angle bisectors of $\backslash angle\; ABC,\; \backslash angle\; BCA,\; \backslash text\; \backslash angle\; BAC$ meet. The distance from vertex $A$ to the incenter $I$ is: :$d(A,\; I)\; =\; c\; \backslash frac\; =\; b\; \backslash frac.$

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 areEncyclopedia of Triangle Centers

, accessed 2014-10-28. :$\backslash \; 1\; :\; 1\; :\; 1.$

Barycentric coordinates

The 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 :$\backslash \; a\; :\; b\; :\; c$ where $a$, $b$, and $c$ are the lengths of the sides of the triangle, or equivalently (using the law of sines) by :$\backslash sin(A):\backslash sin(B):\backslash sin(C)$ where $A$, $B$, and $C$ 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 $(x\_a,y\_a)$, $(x\_b,y\_b)$, and $(x\_c,y\_c)$, and the sides opposite these vertices have corresponding lengths $a$, $b$, and $c$, then the incenter is at :$\backslash left(\backslash frac,\; \backslash frac\backslash right)\; =\; \backslash frac.$

Radius

The inradius $r$ of the incircle in a triangle with sides of length $a$'', $b$'', $c$ is given by :$r\; =\; \backslash sqrt,$ where $s\; =\; (a\; +\; b\; +\; c)/2.$ See Heron's formula.

Distances to the vertices

Denoting the incenter of $\backslash triangle\; ABC$ as $I$, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation :$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 1.$ Additionally, :$IA\; \backslash cdot\; IB\; \backslash cdot\; IC\; =\; 4Rr^2,$ where $R$ and $r$ are the triangle's circumradius and inradius respectively.

Other properties

The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element.

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:''Mathematical Gazette'', July 2003, 323-324. :$d\backslash left(A,\; T\_B\backslash right)\; =\; d\backslash left(A,\; T\_C\backslash right)\; =\; \backslash frac(b\; +\; c\; -\; a).$

Other properties

Suppose the tangency points of the incircle divide the sides into lengths of $x$ and $y$, $y$ and $z$, and ''$z$'' and $x$. Then the incircle has the radius :$r\; =\; \backslash sqrt$ and the area of the triangle is :$\backslash Delta\; =\; \backslash sqrt.$ If the altitudes from sides of lengths $a$'', $b$'', and $c$ are $h\_a$, $h\_b$, and ''$h\_c$'', then the inradius ''$r$'' is one-third of the harmonic mean of these altitudes; that is, :$r\; =\; \backslash frac.$ The product of the incircle radius ''$r$'' and the circumcircle radius $R$ of a triangle with sides $a$'', $b$'', and $c$ isJohnson, Roger A., ''Advanced Euclidean Geometry'', Dover, 2007 (orig. 1929). :$rR\; =\; \backslash frac.$ Some relations among the sides, incircle radius, and circumcircle radius are: :$\backslash begin\; ab\; +\; bc\; +\; ca\; \&=\; s^2\; +\; (4R\; +\; r)r,\; \backslash \backslash \; a^2\; +\; b^2\; +\; c^2\; \&=\; 2s^2\; -\; 2(4R\; +\; r)r.\; \backslash end$ 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. Denoting the center of the incircle of $\backslash triangle\; ABC$ as $I$, we have :$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 1$ and :$IA\; \backslash cdot\; IB\; \backslash cdot\; IC\; =\; 4Rr^2.$ The incircle radius is no greater than one-ninth the sum of the altitudes. The squared distance from the incenter $I$ to the circumcenter $O$ is given by. :$OI^2\; =\; R(R\; -\; 2r)$, and the distance from the incenter to the center $N$ of the nine point circle is : $IN\; =\; \backslash frac(R\; -\; 2r)\; <\; \backslash fracR.$ The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).

Relation to area of the triangle

The radius of the incircle is related to the area of the triangle. The ratio of the area of the incircle to the area of the triangle is less than or equal to $\backslash tfrac$, with equality holding only for equilateral triangles. Suppose $\backslash triangle\; ABC$ has an incircle with radius $r$ and center $I$. Let $a$ be the length of $BC$, $b$ the length of $AC$, and $c$ the length of $AB$''.'' Now, the incircle is tangent to ''$AB$'' at some point $T\_C$, and so $\backslash angle\; AT\_CI$ is right. Thus, the radius $T\_CI$ is an altitude of $\backslash triangle\; IAB$. Therefore, $\backslash triangle\; IAB$ has base length ''$c$'' and height $r$, and so has area $\backslash tfraccr$. Similarly, $\backslash triangle\; IAC$ has area $\backslash tfracbr$ and $\backslash triangle\; IBC$ has area $\backslash tfracar$. Since these three triangles decompose $\backslash triangle\; ABC$, we see that the area $\backslash Delta\; \backslash text\; \backslash triangle\; ABC$ is: :$\backslash Delta\; =\; \backslash frac\; (a\; +\; b\; +\; c)r\; =\; sr,$ and $r\; =\; \backslash frac,$ where $\backslash Delta$ is the area of $\backslash triangle\; ABC$ and $s\; =\; \backslash tfrac(a\; +\; b\; +\; c)$ is its semiperimeter. For an alternative formula, consider $\backslash triangle\; IT\_CA$. This is a right-angled triangle with one side equal to ''$r$'' and the other side equal to $r\; \backslash cot\backslash left(\backslash frac\backslash right)$. The same is true for $\backslash triangle\; IB\text{'}A$. The large triangle is composed of six such triangles and the total area is: :$\backslash Delta\; =\; r^2\; \backslash left(\backslash cot\backslash left(\backslash frac\backslash right)\; +\; \backslash cot\backslash left(\backslash frac\backslash right)\; +\; \backslash cot\backslash left(\backslash frac\backslash right)\backslash right).$

Gergonne triangle and point

The Gergonne triangle (of ''$\backslash triangle\; ABC$'') is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite $A$ is denoted ''$T\_A$'', etc. This Gergonne triangle, ''$\backslash triangle\; T\_AT\_BT\_C$'', is also known as the contact triangle or intouch triangle of ''$\backslash triangle\; ABC$''. Its area is :$K\_T\; =\; K\backslash frac$ where $K$, $r$, and $s$ are the area, radius of the incircle, and semiperimeter of the original triangle, and $a$, $b$, and $c$ are the side lengths of the original triangle. This is the same area as that of the extouch triangle. The three lines ''$AT\_A$'', ''$BT\_B$'' and ''$CT\_C$'' intersect in a single point called the Gergonne point, denoted as ''$G\_e$'' (or triangle center ''X''

Excircles and excenters

$I$), excircles, excenters ($J\_A$, $J\_B$, $J\_C$), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle. An excircle or escribed circle 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. The center of an excircle is the intersection of the internal bisector of one angle (at vertex $A$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex $A$, or the excenter of $A$. 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.

Trilinear coordinates of excenters

While the incenter of ''$\backslash triangle\; ABC$'' has trilinear coordinates $1\; :\; 1\; :\; 1$, the excenters have trilinears $-1\; :\; 1\; :\; 1$, $1\; :\; -1\; :\; 1$, and $1\; :\; 1\; :\; -1$.

Exradii

The radii of the excircles are called the exradii. The exradius of the excircle opposite $A$ (so touching $BC$, centered at $J\_A$) is : $r\_a\; =\; \backslash frac\; =\; \backslash sqrt,$ where $s\; =\; \backslash tfrac(a\; +\; b\; +\; c).$ See Heron's formula.

Derivation of exradii formula

Let the excircle at side $AB$ touch at side $AC$ extended at $G$, and let this excircle's radius be $r\_c$ and its center be $J\_c$. Then $J\_c\; G$ is an altitude of $\backslash triangle\; ACJ\_c$, so $\backslash triangle\; ACJ\_c$ has area $\backslash tfracbr\_c$. By a similar argument, $\backslash triangle\; BCJ\_c$ has area $\backslash tfracar\_c$ and $\backslash triangle\; ABJ\_c$ has area $\backslash tfraccr\_c$. Thus the area $\backslash Delta$ of triangle $\backslash triangle\; ABC$ is : $\backslash Delta\; =\; \backslash frac(a\; +\; b\; -\; c)r\_c\; =\; (s\; -\; c)r\_c$. So, by symmetry, denoting $r$ as the radius of the incircle, : $\backslash Delta\; =\; sr\; =\; (s\; -\; a)r\_a\; =\; (s\; -\; b)r\_b\; =\; (s\; -\; c)r\_c$. By the Law of Cosines, we have : $\backslash cos(A)\; =\; \backslash frac$ Combining this with the identity $\backslash sin^2\; A\; +\; \backslash cos^2\; A\; =\; 1$, we have : $\backslash sin(A)\; =\; \backslash frac$ But $\backslash Delta\; =\; \backslash tfracbc\; \backslash sin(A)$, and so :$\backslash begin\; \backslash Delta\; \&=\; \backslash frac\; \backslash sqrt\; \backslash \backslash \; \&=\; \backslash frac\; \backslash sqrt\; \backslash \backslash \; \&\; =\; \backslash sqrt,\; \backslash end$ which is Heron's formula. Combining this with $sr\; =\; \backslash Delta$, we have :$r^2\; =\; \backslash frac\; =\; \backslash frac.$ Similarly, $(s\; -\; a)r\_a\; =\; \backslash Delta$ gives :$r\_a^2\; =\; \backslash frac$ and :$r\_a\; =\; \backslash sqrt.$

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: :$\backslash Delta\; =\; \backslash sqrt.$

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle. The radius of this Apollonius circle is $\backslash tfrac$ where $r$ is the incircle radius and $s$ is the semiperimeter of the triangle. The following relations hold among the inradius ''$r$'', the circumradius $R$, the semiperimeter ''$s$'', and the excircle radii ''$r\_a$'', ''$r\_b$'', ''$r\_c$'':Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342.

/ref> :$\backslash begin\; r\_a\; +\; r\_b\; +\; r\_c\; \&=\; 4R\; +\; r,\; \backslash \backslash \; r\_a\; r\_b\; +\; r\_b\; r\_c\; +\; r\_c\; r\_a\; \&=\; s^2,\; \backslash \backslash \; r\_a^2\; +\; r\_b^2\; +\; r\_c^2\; \&=\; \backslash left(4R\; +\; r\backslash right)^2\; -\; 2s^2.\; \backslash end$ The circle through the centers of the three excircles has radius $2R$. If ''$H$'' is the orthocenter of ''$\backslash triangle\; ABC$'', then :$\backslash begin\; r\_a\; +\; r\_b\; +\; r\_c\; +\; r\; \&=\; AH\; +\; BH\; +\; CH\; +\; 2R,\; \backslash \backslash \; r\_a^2\; +\; r\_b^2\; +\; r\_c^2\; +\; r^2\; \&=\; AH^2\; +\; BH^2\; +\; CH^2\; +\; (2R)^2.\; \backslash end$

Related constructions

Nine-point circle and Feuerbach point

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 points are: * The midpoint of each side of the triangle * The foot of each altitude * The midpoint of the line segment from each 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 to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that: :... 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 ... 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 ''$\backslash triangle\; ABC$'' with the segments ''$BC$, $CA$,'' and ''$AB$'' are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle are given by * $\backslash \; \backslash left(\; \backslash text\; \backslash ,\; A\backslash right)\; =\; 0\; :\; 1\; :\; 1$ * $\backslash \; \backslash left(\; \backslash text\; \backslash ,\; B\backslash right)\; =\; 1\; :\; 0\; :\; 1$ *$\backslash \; \backslash left(\; \backslash text\; \backslash ,\; C\backslash right)\; =\; 1\; :\; 1\; :\; 0.$ 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 of page). Trilinear coordinates for the vertices of the excentral triangle are given by * $(\backslash text\; \backslash ,\; A)\; =\; -1\; :\; 1\; :\; 1$ * $(\backslash text\; \backslash ,\; B)\; =\; 1\; :\; -1\; :\; 1$ *$(\backslash text\; \backslash ,\; C)\; =\; 1\; :\; 1\; :\; -1.$

Equations for four circles

Let ''$x:y:z$'' be a variable point in trilinear coordinates, and let ''$u=\backslash cos^2\backslash left\; (\; A/2\; \backslash right\; )$'', ''$v=\backslash cos^2\backslash left\; (\; B/2\; \backslash right\; )$'', ''$w=\backslash cos^2\backslash left\; (\; C/2\; \backslash right\; )$''. The four circles described above are given equivalently by either of the two given equations:Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books * Incircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; -\; 2vwyz\; -\; 2wuzx\; -\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$ *''$A$-''excircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; -\; 2vwyz\; +\; 2wuzx\; +\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$ * ''$B$-''excircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; +\; 2vwyz\; -\; 2wuzx\; +\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$ *''$C$-''excircle: *:$\backslash begin\; u^2\; x^2\; +\; v^2\; y^2\; +\; w^2\; z^2\; +\; 2vwyz\; +\; 2wuzx\; -\; 2uvxy\; \&=\; 0\; \backslash \backslash \; \backslash pm\backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \backslash pm\; \backslash sqrt\backslash cos\backslash left(\backslash frac\backslash right)\; \&=\; 0\; \backslash end$

Euler's theorem

Euler's theorem states that in a triangle: :$(R\; -\; r)^2\; =\; d^2\; +\; r^2,$ where ''$R$'' and ''$r$'' are the circumradius and inradius respectively, and ''$d$'' is the distance between the circumcenter and the incenter. For excircles the equation is similar: :$\backslash left(R\; +\; r\_\backslash text\backslash right)^2\; =\; d\_\backslash text^2\; +\; r\_\backslash text^2,$ where ''$r\_\backslash text$'' is the radius of one of the excircles, and ''$d\_\backslash text$'' is the distance between the circumcenter and that excircle's center.Nelson, Roger, "Euler's triangle inequality via proof without words," ''Mathematics Magazine'' 81(1), February 2008, 58-61.

/ref>

Generalization to other polygons

Some (but not all) quadrilaterals have an incircle. These are called tangential quadrilaterals. 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. 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

* * * * * * * * * *

Notes

References

* * * *

External links

Derivation of formula for radius of incircle of a triangle

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Interactive

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nbsp; With interactive animations

An interactive animated demonstration

Equal Incircles Theorem

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Five Incircles Theorem

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Pairs of Incircles in a Quadrilateral

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An interactive Java applet for the incenter

{{DEFAULTSORT:Incircle And Excircles Of A Triangle Category:Circles defined for a triangle