# Incircle and excircles of a triangle

A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (JA,JB,JC), internal angle bisectors (red) and external angle bisectors (green). 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.[1]

An excircle or escribed circle[2] 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.[3]

The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors.[3][4] 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.[3] 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.[5]:p. 182

Polygons with more than three sides do not all have an incircle tangent to all sides; those that do are called tangential polygons. See also Tangent lines to circles.

## Incircle and incenter

A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔTaTbTc) and Gergonne point (green, Ge)

Suppose ${\displaystyle \triangle ABC}$ 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 And let ${\displaystyle T_{a},T_{b},{\text{ and }}T_{c}}$ be the touchpoints where the incircle touches BC , AC and AB

### Incenter

The incenter is the point where the internal angle bisectors of ${\displaystyle \angle ABC,\angle BCA,{\text{ and }}\angle BAC}$ meet.

The distance from vertex A to the incenter I is:

${\displaystyle d(A,I)=c{\frac {\sin({\frac {1}{2}}\angle B)}{\cos({\frac {1}{2}}\angle C)}}=b{\frac {\sin({\frac {1}{2}}\angle C)}{\cos({\frac {1}{2}}\angle B)}}}$

#### Trilinear coordinates of the incenter

The trilinear coordinates for a point in the triangle is the ratio of distances to the triangle sides. Because the Incenter is the same distance of all sides the trilinear coordinates for the incenter are [6]

${\displaystyle \ 1:1:1.}$

#### Barycentric coordinates of the incenter

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

${\displaystyle \ a:b:c}$

where ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are the lengths of the sides of the triangle, or equivalently (using the law of sines) by

${\displaystyle \sin(A):\sin(B):\sin(C)}$

where ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ are the angles at the three vertices.

#### Cartesian coordinates of the incenter

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 ${\displaystyle (x_{a},y_{a})}$, ${\displaystyle (x_{b},y_{b})}$, and ${\displaystyle (x_{c},y_{c})}$, and the sides opposite these vertices have corresponding lengths ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, then the incenter is at

${\displaystyle {\bigg (}{\frac {ax_{a}+bx_{b}+cx_{c}}{a+b+c}},{\frac {ay_{a}+by_{b}+cy_{c}}{a+b+c}}{\bigg )}={\frac {a(x_{a},y_{a})+b(x_{b},y_{b})+c(x_{c},y_{c})}{a+b+c}}.}$

#### Distances from the incenter to the 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[7]

${\displaystyle {\frac {IA\cdot IA}{CA\cdot AB}}+{\frac {IB\cdot IB}{AB\cdot BC}}+{\frac {IC\cdot IC}{BC\cdot CA}}=1.}$

${\displaystyle IA\cdot IB\cdot IC=4Rr^{2},}$

#### Other incenter 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.[6]

### Incircle and its radius properties

#### Distances between vertex and nearest touchpoints

The distances from a vertex to the two nearest touchpoints are equal:

${\displaystyle d(A,T_{B})=d(A,T_{C}))={\frac {1}{2}}(b+c-a)}$

#### Other incircle 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[9]

${\displaystyle r={\sqrt {\frac {xyz}{x+y+z}}}}$

and the area of the triangle is

${\displaystyle \Delta ={\sqrt {xyz(x+y+z)}}.}$

If the altitudes from sides of lengths a, b, and c are ha, hb, and hc then the inradius r is one-third of the harmonic mean of these altitudes, i.e.

${\displaystyle r={\frac {1}{h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1}}}.}$

The product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is[5]:p. 189, #298(d)

${\displaystyle rR={\frac {abc}{2(a+b+c)}}.}$

${\displaystyle ab+bc+ca=s^{2}+(4R+r)r,}$
${\displaystyle a^{2}+b^{2}+c^{2}=2s^{2}-2(4R+r)r.}$

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.[11]

Denoting the center of the incircle of triangle ABC as I, we have[12]

${\displaystyle {\frac {IA\cdot IA}{CA\cdot AB}}+{\frac {IB\cdot IB}{AB\cdot BC}}+{\frac {IC\cdot IC}{BC\cdot CA}}=1}$

and[13]:p.121,#84

${\displaystyle IA\cdot IB\cdot IC=4Rr^{2}.}$

The distance from any vertex to the incircle tangency on either adjacent side is half the sum of the vertex's adjacent sides minus half the opposite side.[14] Thus for example for vertex B and adjacent tangencies TA and TC,

${\displaystyle BT_{A}=BT_{C}={\frac {BC+AB-AC}{2}}.}$

The incircle radius is no greater than one-ninth the sum of the altitudes.[15]:p. 289

The squared distance from the incenter I to the circumcenter O is given by[16]:p.232

${\displaystyle OI^{2}=R(R-2r),}$

and the distance from the incenter to the center N of the nine point circle is[16]:p.232

${\displaystyle IN={\frac {1}{2}}(R-2r)<{\frac {1}{2}}R.}$

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).[16]:p.233, Lemma 1

#### Relation to area of the triangle

A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔTaTbTc) and Gergonne point (green, Ge)

The radii of the incircle is related to the area of the triangle.[17] The ratio of the area of the incircle to the area of the triangle is less than or equal to ${\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}$, with equality holding only for equilateral triangles.[18]

Suppose ${\displaystyle \triangle ABC}$ 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 C′, and so ${\displaystyle \angle AT_{c}I}$ is right. Thus the radius TcI is an altitude of ${\displaystyle \triangle IAB}$. Therefore, ${\displaystyle \triangle IAB}$ has base length c and height r, and so has area ${\displaystyle {\tfrac {1}{2}}cr}$. Similarly, ${\displaystyle \triangle IAC}$ has area ${\displaystyle {\tfrac {1}{2}}br}$ and ${\displaystyle \triangle IBC}$ has area ${\displaystyle {\tfrac {1}{2}}ar}$. Since these three triangles decompose ${\displaystyle \triangle ABC}$, we see that the area ${\displaystyle \Delta {\text{ of }}\triangle ABC}$ is:

${\displaystyle \Delta ={\frac {1}{2}}(a+b+c)r=sr,}$      and      ${\displaystyle r={\frac {\Delta }{s}},}$

where ${\displaystyle \Delta }$ is the area of ${\displaystyle \triangle ABC}$ and ${\displaystyle s={\frac {1}{2}}(a+b+c)}$ is its semiperimeter.

For an alternative formula, consider ${\displaystyle \triangle IT_{c}A}$. This is a right-angled triangle with one side equal to r and the other side equal to ${\displaystyle r\cot {\frac {\angle A}{2}}}$. The same is true for ${\displaystyle \triangle IB'A}$. The large triangle is composed of 6 such triangles and the total area is:

${\displaystyle \Delta =r^{2}\cdot (\cot {\frac {\angle A}{2}}+\cot {\frac {\angle B}{2}}+\cot {\frac {\angle C}{2}})}$

### Gergonne triangle and point

A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔTaTbTc) and Gergonne point (green, Ge)

The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. The touchpoint opposite A is denoted TA, etc.

This Gergonne triangle TATBTC is also known as the contact triangle or intouch triangle of ABC. Its area is

${\displaystyle K_{T}=K{\frac {2r^{2}s}{abc}}}$

where ${\displaystyle K}$, ${\displaystyle r}$, ${\displaystyle s}$ are the area, radius of the incircle and semiperimeter of the original triangle, and ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ are the side lengths of the original triangle. This is the same area as that of the extouch triangle.[19]

The three lines ATA, BTB and CTC intersect in a single point called the Gergonne point, denoted as Ge - X(7). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.[20]

Interestingly, the Gergonne point of a triangle is the symmedian point of the Gergonne triangle. For a full set of properties of the Gergonne point see.[21]

Trilinear coordinates for the vertices of the intouch triangle are given by

• ${\displaystyle {\text{vertex}}\,T_{A}=0:\sec ^{2}\left({\frac {B}{2}}\right):\sec ^{2}\left({\frac {C}{2}}\right)}$
• ${\displaystyle {\text{vertex}}\,T_{B}=\sec ^{2}\left({\frac {A}{2}}\right):0:\sec ^{2}\left({\frac {C}{2}}\right)}$
• ${\displaystyle {\text{vertex}}\,T_{C}=\sec ^{2}\left({\frac {A}{2}}\right):\sec ^{2}\left({\frac {B}{2}}\right):0}$

Trilinear coordinates for the Gergonne point are given by

${\displaystyle \sec ^{2}\left({\frac {A}{2}}\right):\sec ^{2}\left({\frac {B}{2}}\right):\sec ^{2}\left({\frac {C}{2}}\right)}$,

or, equivalently, by the Law of Sines,

${\displaystyle {\frac {bc}{b+c-a}}:{\frac {ca}{c+a-b}}:{\frac {ab}{a+b-c}}}$.

## Excircles and excenters

A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (JA,JB,JC), internal angle bisectors (red) and external angle bisectors (green). The green triangle is the excentral triangle.

An excircle or escribed circle[22] 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.[3]

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.[3] 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.[5]:p. 182

### Trilinear coordinates of excenters

While A triangle's incenter has trilinear coordinates${\displaystyle 1:1:1.}$ Its excenters (the centers of its excircles) have trilinears ${\displaystyle -1:1:1}$ , ${\displaystyle 1:-1:1}$ and ${\displaystyle 1:1:-1.}$

the exradius of the Excircle opposite A (so touching BC, centered at ${\displaystyle J_{A}}$ is

${\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.}$[23]

where ${\displaystyle s={\frac {1}{2}}(a+b+c)}$

Let the excircle at side AB touch at side AC extended at G, and let this excircle's radius be ${\displaystyle r_{c}}$ and its center be ${\displaystyle J_{c}}$.

Then ${\displaystyle J_{c}G}$ is an altitude of ${\displaystyle \triangle ACJ_{c}}$, so ${\displaystyle \triangle ACJ_{c}}$ has area ${\displaystyle {\tfrac {1}{2}}br_{c}}$. By a similar argument,${\displaystyle \triangle BCJ_{c}}$ has area ${\displaystyle {\tfrac {1}{2}}ar_{c}}$ and ${\displaystyle \triangle ABJ_{c}}$ has area ${\displaystyle {\tfrac {1}{2}}cr_{c}}$. Thus the area ${\displaystyle \Delta }$ of triangle ${\displaystyle \triangle ABC}$ is

${\displaystyle \Delta ={\frac {1}{2}}(a+b-c)r_{c}=(s-c)r_{c}}$.

So, by symmetry, denoting ${\displaystyle r}$ as the radius of the incircle,

${\displaystyle \Delta =sr=(s-a)r_{a}=(s-b)r_{b}=(s-c)r_{c}}$.

By the Law of Cosines, we have

${\displaystyle \cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}}$

Combining this with the identity ${\displaystyle \sin ^{2}A+\cos ^{2}A=1}$, we have

${\displaystyle \sin A={\frac {\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}{2bc}}}$

But ${\displaystyle \Delta ={\tfrac {1}{2}}bc\sin A}$, and so

{\displaystyle {\begin{aligned}\Delta &={\frac {1}{4}}{\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}\\&={\frac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {s(s-a)(s-b)(s-c)}},\end{aligned}}}

which is Heron's formula.

Combining this with ${\displaystyle sr=\Delta }$, we have

${\displaystyle r^{2}={\frac {\Delta ^{2}}{s^{2}}}={\frac {(s-a)(s-b)(s-c)}{s}}.}$

Similarly, ${\displaystyle (s-a)r_{a}=\Delta }$ gives

${\displaystyle r_{a}^{2}={\frac {s(s-b)(s-c)}{s-a}}}$

and

${\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.}$[24]

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:[25]

${\displaystyle \Delta ={\sqrt {rr_{a}r_{b}r_{c}}}.}$

The ratio of the area of the incircle to the area of the triangle is less than or equal to ${\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}$, with equality holding only for equilateral triangles.[26]

### Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle.[27] The radius of this Apollonius circle is ${\displaystyle {\frac {r^{2}+s^{2}}{4r}}}$ where r is the incircle radius and s is the semiperimeter of the triangle.[28]

The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii ra, rb, rc:[10]

${\displaystyle r_{a}+r_{b}+r_{c}=4R+r,}$
${\displaystyle r_{a}r_{b}+r_{b}r_{c}+r_{c}r_{a}=s^{2},}$
${\displaystyle r_{a}^{2}+r_{b}^{2}+r_{c}^{2}=(4R+r)^{2}-2s^{2},}$

The circle through the centers of the three excircles has radius 2R.[10]

If H is the orthocenter of triangle ABC, then[10]

${\displaystyle r_{a}+r_{b}+r_{c}+r=AH+BH+CH+2R,}$
${\displaystyle r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=AH^{2}+BH^{2}+CH^{2}+(2R)^{2}.}$

### Nagel triangle and Nagel point

The extouch triangle (ΔTATBTC, with red boundary) and the Nagel point (blue, N) of a triangle (ΔABC, with black boundary). The orange circles are the excircles of the triangle.

The Nagel triangle or extouch triangle of ABC is denoted by the vertices TA, TB and TC that are the three points where the excircles touch the reference triangle ABC and where TA is opposite of A, etc. This triangle TATBTC is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle TATBTC is called the Mandart circle.

The three lines ATA, BTB and CTC are called the splitters of the triangle; they each bisect the perimeter of the triangle,

${\displaystyle AB+BT_{A}=AC+CT_{A}={\frac {1}{2}}\left(AB+BC+AC\right)}$

The splitters intersect in a single point, the triangle's Nagel point Na - X(8).

Trilinear coordinates for the vertices of the extouch triangle are given by

• ${\displaystyle {\text{vertex}}\,T_{A}=0:\csc ^{2}\left({\frac {B}{2}}\right):\csc ^{2}\left({\frac {C}{2}}\right)}$
• ${\displaystyle {\text{vertex}}\,T_{B}=\csc ^{2}\left({\frac {A}{2}}\right):0:\csc ^{2}\left({\frac {C}{2}}\right)}$
• ${\displaystyle {\text{vertex}}\,T_{C}=\csc ^{2}\left({\frac {A}{2}}\right):\csc ^{2}\left({\frac {B}{2}}\right):0}$

Trilinear coordinates for the Nagel point are given by

${\displaystyle \csc ^{2}\left({\frac {A}{2}}\right):\csc ^{2}\left({\frac {B}{2}}\right):\csc ^{2}\left({\frac {C}{2}}\right)}$,

or, equivalently, by the Law of Sines,

${\displaystyle {\frac {b+c-a}{a}}:{\frac {c+a-b}{b}}:{\frac {a+b-c}{c}}}$.

It is the isotomic conjugate of the Gergonne point.

## Relation to area of the triangle

The radii of the incircles and excircles are closely related to the area of the triangle.[29]

### Incircle

Suppose ${\displaystyle \triangle ABC}$ 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 C′, and so ${\displaystyle \angle AC'I}$ is right. Thus the radius C'I is an altitude of ${\displaystyle \triangle IAB}$. Therefore, ${\displaystyle \triangle IAB}$ has base length c and height r, and so has area ${\displaystyle {\tfrac {1}{2}}cr}$. Similarly, ${\displaystyle \triangle IAC}$ has area ${\displaystyle {\tfrac {1}{2}}br}$ and ${\displaystyle \triangle IBC}$ has area ${\displaystyle {\tfrac {1}{2}}ar}$. Since these three triangles decompose ${\displaystyle \triangle ABC}$, we see that

${\displaystyle \Delta ={\frac {1}{2}}(a+b+c)r=sr,}$      and      ${\displaystyle r={\frac {\Delta }{s}},}$

where ${\displaystyle \Delta }$ is the area of ${\displaystyle \triangle ABC}$ and ${\displaystyle s={\frac {1}{2}}(a+b+c)}$ is its semiperimeter.

For an alternative formula, consider ${\displaystyle \triangle IC'A}$. This is a right-angled triangle with one side equal to r and the other side equal to ${\displaystyle r\cot {\frac {\angle A}{2}}}$. The same is true for ${\displaystyle \triangle IB'A}$. The large triangle is composed of 6 such triangles and the total area is:

${\displaystyle \Delta =r^{2}\cdot (\cot {\frac {\angle A}{2}}+\cot {\frac {\angle B}{2}}+\cot {\frac {\angle C}{2}})}$

### Excircles

The radii of the excircles are called the exradii. Let the excircle at side AB touch at side AC extended at G, and let this excircle's radius be ${\displaystyle r_{c}}$ and its center be ${\displaystyle I_{c}}$. Then ${\displaystyle I_{c}G}$ is an altitude of ${\displaystyle \triangle ACI_{c}}$, so ${\displaystyle \triangle ACI_{c}}$ has area ${\displaystyle {\tfrac {1}{2}}br_{c}}$. By a similar argument, ${\displaystyle \triangle BCI_{c}}$ has area ${\displaystyle {\tfrac {1}{2}}ar_{c}}$ and ${\displaystyle \triangle ABI_{c}}$ has area ${\displaystyle {\tfrac {1}{2}}cr_{c}}$. Thus

${\displaystyle \Delta ={\frac {1}{2}}(a+b-c)r_{c}=(s-c)r_{c}}$.

So, by symmetry,

${\displaystyle \Delta =sr=(s-a)r_{a}=(s-b)r_{b}=(s-c)r_{c}}$.

By the Law of Cosines, we have

${\displaystyle \cos A={\frac {b^{2}+c^{2}-a^{2}}{2bc}}}$

Combining this with the identity ${\displaystyle \sin ^{2}A+\cos ^{2}A=1}$, we have

${\displaystyle \sin A={\frac {\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}{2bc}}}$

But ${\displaystyle \Delta ={\tfrac {1}{2}}bc\sin A}$, and so

{\displaystyle {\begin{aligned}\Delta &={\frac {1}{4}}{\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}\\&={\frac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {s(s-a)(s-b)(s-c)}},\end{aligned}}}

which is Heron's formula.

Combining this with ${\displaystyle sr=\Delta }$, we have

${\displaystyle r^{2}={\frac {\Delta ^{2}}{s^{2}}}={\frac {(s-a)(s-b)(s-c)}{s}}.}$

Similarly, ${\displaystyle (s-a)r_{a}=\Delta }$ gives

${\displaystyle r_{a}^{2}={\frac {s(s-b)(s-c)}{s-a}}}$

and

${\displaystyle r_{a}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}}.}$[30]

From these formulas 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:[31]

${\displaystyle \Delta ={\sqrt {rr_{a}r_{b}r_{c}}}.}$

The ratio of the area of the incircle to the area of the triangle is less than or equal to ${\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}$, with equality holding only for equilateral triangles.[32]

## Related constructions

### Nine-point circle and Feuerbach point

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 points are:

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... (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 ABC with the segments BC, CA, AB are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle are given by

• ${\displaystyle \ ({\text{vertex opposite}}\,A)=0:1:1}$
• ${\displaystyle \ ({\text{vertex opposite}}\,B)=1:0:1}$
• ${\displaystyle \ ({\text{vertex opposite}}\,C)=1:1:0}$

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

• ${\displaystyle ({\text{vertex opposite}}\,A)=-1:1:1}$
• ${\displaystyle ({\text{vertex opposite}}\,B)=1:-1:1}$
• ${\displaystyle ({\text{vertex opposite}}\,C)=1:1:-1}$

## Equations for four circles

Let x : y : z be a variable point in trilinear coordinates, and let u = cos2(A/2), v = cos2(B/2), w = cos2(C/2). The four circles described above are given equivalently by either of the two given equations:[35]:p. 210–215

• Incircle:
${\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0}$
${\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}$
• A-excircle:
${\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz+2wuzx+2uvxy=0}$
${\displaystyle \pm {\sqrt {-x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}$
• B-excircle:
${\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz-2wuzx+2uvxy=0}$
${\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {-y}}\cos {\frac {B}{2}}\pm {\sqrt {z}}\cos {\frac {C}{2}}=0}$
• C-excircle:
${\displaystyle \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz+2wuzx-2uvxy=0}$
${\displaystyle \pm {\sqrt {x}}\cos {\frac {A}{2}}\pm {\sqrt {y}}\cos {\frac {B}{2}}\pm {\sqrt {-z}}\cos {\frac {C}{2}}=0}$

## Euler's theorem

Euler's theorem states that in a triangle:

${\displaystyle (R-r)^{2}=d^{2}+r^{2},}$

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:

${\displaystyle (R+r_{ex})^{2}=d_{ex}^{2}+r_{ex}^{2},}$

where rex is the radius of one of the excircles, and dex is the distance between the circumcenter and this excircle's center.[36][37][38]

## 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—one that is tangent to each side—is called a tangential polygon.

## Notes

1. ^ Kay (1969, p. 140)
2. ^ Altshiller-Court (1925, p. 74)
3. Altshiller-Court (1925, p. 73)
4. ^ Kay (1969, p. 117)
5. ^ a b c Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929).
6. ^ a b Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine., accessed 2014-10-28.
7. ^ Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette, 96: 161–165.
8. ^ Altshiller-Court, Nathan (1980), College Geometry, Dover Publications. #84, p. 121.
9. ^ Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
10. ^ a b c d Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
11. ^ Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
12. ^ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
13. ^ Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
14. ^ Mathematical Gazette, July 2003, 323-324.
15. ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
16. ^ a b c Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. MR 2877263..
17. ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
18. ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
19. ^ Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
20. ^ 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
21. ^ Dekov, Deko (2009). "Computer-generated Mathematics : The Gergonne Point" (PDF). Journal of Computer-generated Euclidean Geometry. 1: 1–14.
22. ^ Altshiller-Court (1925, p. 74)
23. ^ Altshiller-Court (1925, p. 79)
24. ^ Altshiller-Court (1925, p. 79)
25. ^ 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.)
26. ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
27. ^ Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.
28. ^ Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
29. ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
30. ^ Altshiller-Court (1925, p. 79)
31. ^ 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.)
32. ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
33. ^ Altshiller-Court (1925, pp. 103–110)
34. ^ Kay (1969, pp. 18,245)
35. ^ 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
36. ^ Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
37. ^ Johnson, R. A. Modern Geometry, Houghton Mifflin, Boston, 1929: p. 187.
38. ^ Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140.

## References

• Altshiller-Court, Nathan (1925), College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), New York: Barnes & Noble, LCCN 52013504
• Kay, David C. (1969), College Geometry, New York: Holt, Rinehart and Winston, LCCN 69012075
• Kimberling, Clark (1998). "Triangle Centers and Central Triangles". Congressus Numerantium (129): i–xxv,1–295.
• Kiss, Sándor (2006). "The Orthic-of-Intouch and Intouch-of-Orthic Triangles". Forum Geometricorum (6): 171–177.