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Incircle and excircles

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(Redirected fromIncircle)

Circles tangent to all three sides of a triangle

"Incircle" redirects here. For incircles of non-triangle polygons, seeTangential quadrilateral andTangential polygon.

Not to be confused withCircumcircle.

Incircle and excircles of a triangle.
  Extended sides of triangle△ABC
  Incircle (incenter atI)
  Excircles (excenters atJ_A,J_B,J_C)
  Internalangle bisectors
  External angle bisectors (forming the excentral triangle)

Ingeometry, theincircle orinscribed circle of atriangle is the largestcircle that can be contained in the triangle; it touches (istangent to) the three sides. The center of the incircle is atriangle center called the triangle'sincenter.^[1]

Anexcircle orescribed circle^[2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to theextensions 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 theincenter, can be found as the intersection of the threeinternal angle bisectors.^[3]^[4] The center of an excircle is the intersection of the internal bisector of one angle (at vertexA, for example) and theexternal bisectors of the other two. The center of this excircle is called theexcenter relative to the vertexA, or theexcenter ofA.^[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 anorthocentric system.^[5]

Incircle and Incenter

See also:Incenter

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

Also let $T_{A}$ , $T_{B}$ , and $T_{C}$ be the touchpoints where the incircle touches ${\overline {BC}}$ , ${\overline {AC}}$ , and ${\overline {AB}}$ .

Incenter

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

Trilinear coordinates

Thetrilinear 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^[6]

\ 1:1:1.

Barycentric coordinates

Thebarycentric 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

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

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

\sin A:\sin B:\sin C

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

Cartesian coordinates

TheCartesian 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 $a {\displaystyle a}$ , $b {\displaystyle b}$ , and $c {\displaystyle c}$ , then the incenter is at^{[citation needed]}

\left({\frac {ax_{a}+bx_{b}+cx_{c}}{a+b+c}},{\frac {ay_{a}+by_{b}+cy_{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}}.

Radius

The inradius $r {\displaystyle r}$ of the incircle in a triangle with sides of length $a {\displaystyle a}$ , $b {\displaystyle b}$ , $c {\displaystyle c}$ is given by^[7]

r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}},

where $s={\tfrac {1}{2}}(a+b+c)$ is thesemiperimeter (seeHeron's formula).

The tangency points of the incircle divide the sides into segments of lengths $s-a$ from $A {\displaystyle A}$ , $s-b$ from $B {\displaystyle B}$ , and $s-c$ from $C {\displaystyle C}$ (seeTangent lines to a circle).^[8]

Distances to the vertices

Denote the incenter of $\triangle ABC$ as $I {\displaystyle I}$ .

The distance from vertex $A {\displaystyle A}$ to the incenter $I {\displaystyle I}$ is:

{\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}}}}.

Derivation of the formula stated above

Use theLaw of sines in the triangle $\triangle IAB$ .

We get ${\frac {\overline {AI}}{\sin {\frac {B}{2}}}}={\frac {c}{\sin \angle AIB}}$ .We have that $\angle AIB=\pi -{\frac {A}{2}}-{\frac {B}{2}}={\frac {\pi }{2}}+{\frac {C}{2}}$ .

It follows that ${\overline {AI}}=c\ {\frac {\sin {\frac {B}{2}}}{\cos {\frac {C}{2}}}}$ .

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^[9]

{\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.

Additionally,^[10]

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

where $R {\displaystyle R}$ and $r {\displaystyle r}$ are the triangle'scircumradius andinradius respectively.

Other properties

The collection of triangle centers may be given the structure of agroup under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms theidentity 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; for example:^[11]

d\left(A,T_{B}\right)=d\left(A,T_{C}\right)={\tfrac {1}{2}}(b+c-a)=s-a.

Other properties

If thealtitudes from sides of lengths $a {\displaystyle a}$ , $b {\displaystyle b}$ , and $c {\displaystyle c}$ are $h_{a}$ , $h_{b}$ , and $h_{c}$ , then the inradius $r {\displaystyle r}$ is one third theharmonic mean of these altitudes; that is,^[12]

r={\frac {1}{{\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}}.

The product of the incircle radius $r {\displaystyle r}$ and thecircumcircle radius $R {\displaystyle R}$ of a triangle with sides $a {\displaystyle a}$ , $b {\displaystyle b}$ , and $c {\displaystyle c}$ is^[13]

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

Some relations among the sides, incircle radius, and circumcircle radius are:^[14]

{\begin{aligned}ab+bc+ca&=s^{2}+(4R+r)r,\\a^{2}+b^{2}+c^{2}&=2s^{2}-2(4R+r)r.\end{aligned}}

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.^[15]

The incircle radius is no greater than one-ninth the sum of the altitudes.^[16]^: 289

The squared distance from the incenter $I {\displaystyle I}$ to thecircumcenter $O {\displaystyle O}$ is given by^[17]^: 232

{\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]

and the distance from the incenter to the center $N {\displaystyle N}$ of thenine point circle is^[17]^: 232

{\overline {IN}}={\tfrac {1}{2}}(R-2r)<{\tfrac {1}{2}}R.

The incenter lies in themedial triangle (whose vertices are the midpoints of the sides).^[17]^{: 233, Lemma 1}

Relation to area of the triangle

"Inradius" redirects here. For the three-dimensional equivalent, seeInscribed sphere.

The radius of the incircle is related to thearea of the triangle.^[18] The ratio of the area of the incircle to the area of the triangle is less than or equal to $\pi {\big /}3{\sqrt {3}}$ ,with equality holding only forequilateral triangles.^[19]

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

Now, the incircle is tangent to ${\overline {AB}}$ at some point $T_{C}$ , and so $\angle AT_{C}I$ is right. Thus, the radius $T_{C}I$ is analtitude of $\triangle IAB$ .

Therefore, $\triangle IAB$ has base length $c {\displaystyle c}$ and height $r {\displaystyle r}$ , and so has area ${\tfrac {1}{2}}cr$ .

Similarly, $\triangle IAC$ has area ${\tfrac {1}{2}}br$ and $\triangle IBC$ has area ${\tfrac {1}{2}}ar$ .

Since these three triangles decompose $\triangle ABC$ , we see that the area $\Delta {\text{ of}}\triangle ABC$ is:

\Delta ={\tfrac {1}{2}}(a+b+c)r=sr,

and $r={\frac {\Delta }{s}},$

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

For an alternative formula, consider $\triangle IT_{C}A$ . This is a right-angled triangle with one side equal to $r {\displaystyle r}$ and the other side equal to $r\cot {\tfrac {A}{2}}$ . The same is true for $\triangle IB'A$ . The large triangle is composed of six such triangles and the total area is:^{[citation needed]}

\Delta =r^{2}\left(\cot {\tfrac {A}{2}}+\cot {\tfrac {B}{2}}+\cot {\tfrac {C}{2}}\right).

Gergonne triangle and point

Triangle△ABC
  Incircle (incenter atI)
  Contact triangle△T_AT_BT_C
  Lines between opposite vertices of△ABC and△T_AT_BT_C (concur at Gergonne pointG_e)

TheGergonne triangle (of $\triangle ABC$ ) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite $A {\displaystyle A}$ is denoted $T_{A}$ , etc.

This Gergonne triangle, $\triangle T_{A}T_{B}T_{C}$ , is also known as thecontact triangle orintouch triangle of $\triangle ABC$ . Its area is

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

where $K {\displaystyle K}$ , $r {\displaystyle r}$ , and $s {\displaystyle s}$ are the area, radius of the incircle, and semiperimeter of the original triangle, and $a {\displaystyle a}$ , $b {\displaystyle b}$ , and $c {\displaystyle c}$ are the side lengths of the original triangle. This is the same area as that of theextouch triangle.^[20]

The three lines $AT_{A}$ , $BT_{B}$ , and $CT_{C}$ intersect in a single point called theGergonne point, denoted as $G_{e}$ (ortriangle centerX₇). The Gergonne point lies in the openorthocentroidal disk punctured at its own center, and can be any point therein.^[21]

The Gergonne point of a triangle has a number of properties, including that it is thesymmedian point of the Gergonne triangle.^[22]

Trilinear coordinates for the vertices of the intouch triangle are given by^{[citation needed]}

{\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}}

Trilinear coordinates for the Gergonne point are given by^{[citation needed]}

\sec ^{2}{\tfrac {A}{2}}:\sec ^{2}{\tfrac {B}{2}}:\sec ^{2}{\tfrac {C}{2}},

or, equivalently, by theLaw of Sines,

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

Excircles and excenters

Extended sides of△ABC
  Incircle (incenter atI)
  Excircles (excenters atJ_A,J_B,J_C)
  Internalangle bisectors
  External angle bisectors (forming the excentral triangle)

Anexcircle orescribed circle^[2] of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to theextensions 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 $A {\displaystyle A}$ , for example) and theexternal bisectors of the other two. The center of this excircle is called theexcenter relative to the vertex $A {\displaystyle A}$ , or theexcenter of $A {\displaystyle A}$ .^[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 anorthocentric system.^[5]

Trilinear coordinates of excenters

While theincenter of $\triangle ABC$ hastrilinear coordinates $1:1:1$ , the excenters have trilinears^{[citation needed]}

{\begin{array}{rrcrcr}J_{A}=&-1&:&1&:&1\\J_{B}=&1&:&-1&:&1\\J_{C}=&1&:&1&:&-1\end{array}}

Exradii

The radii of the excircles are called theexradii.

The exradius of the excircle opposite $A {\displaystyle A}$ (so touching $B C {\displaystyle BC}$ , centered at $J_{A}$ ) is^[23]^[24]

r_{a}={\frac {rs}{s-a}}={\sqrt {\frac {s(s-b)(s-c)}{s-a}}},

where

s={\tfrac {1}{2}}(a+b+c).

SeeHeron's formula.

Derivation of exradii formula

Source:^[23]

Let the excircle at side $A B {\displaystyle AB}$ touch at side $A C {\displaystyle AC}$ extended at $G {\displaystyle G}$ , and let this excircle'sradius be $r_{c}$ and its center be $J_{c}$ . Then $J_{c}G$ is an altitude of $\triangle ACJ_{c}$ , so $\triangle ACJ_{c}$ has area ${\tfrac {1}{2}}br_{c}$ . By a similar argument, $\triangle BCJ_{c}$ has area ${\tfrac {1}{2}}ar_{c}$ and $\triangle ABJ_{c}$ has area ${\tfrac {1}{2}}cr_{c}$ . Thus the area $\Delta$ of triangle $\triangle ABC$ is

\Delta ={\tfrac {1}{2}}(a+b-c)r_{c}=(s-c)r_{c}

.

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

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

.

By theLaw of Cosines, we have

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

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

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

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

{\begin{aligned}\Delta &={\tfrac {1}{4}}{\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}\\[5mu]&={\tfrac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\[5mu]&={\sqrt {s(s-a)(s-b)(s-c)}},\end{aligned}}

which isHeron's formula.

Combining this with $sr=\Delta$ , we have

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

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

{\begin{aligned}&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{aligned}}

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:^[25]

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

Other excircle properties

The circularhull of the excircles is internally tangent to each of the excircles and is thus anApollonius circle.^[26] The radius of this Apollonius circle is ${\tfrac {r^{2}+s^{2}}{4r}}$ where $r {\displaystyle r}$ is the incircle radius and $s {\displaystyle s}$ is the semiperimeter of the triangle.^[27]

The following relations hold among the inradius $r {\displaystyle r}$ , the circumradius $R {\displaystyle R}$ , the semiperimeter $s {\displaystyle s}$ , and the excircle radii $r_{a}$ , $r_{b}$ , $r_{c}$ :^[14]

{\begin{aligned}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{aligned}}

The circle through the centers of the three excircles has radius $2R$ .^[14]

If $H {\displaystyle H}$ is theorthocenter of $\triangle ABC$ , then^[14]

{\begin{aligned}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{aligned}}

Nagel triangle and Nagel point

Main article:Extouch triangle

Extended sides of triangle△ABC
  Excircles of△ABC (tangent atT_A. T_B, T_C)
  Nagel/Extouch triangle△T_AT_BT_C
  Splitters: lines connecting opposite vertices of△ABC and△T_AT_BT_C (concur atNagel pointN)

TheNagel triangle orextouch triangle of $\triangle ABC$ is denoted by the vertices $T_{A}$ , $T_{B}$ , and $T_{C}$ that are the three points where the excircles touch the reference $\triangle ABC$ and where $T_{A}$ is opposite of $A {\displaystyle A}$ , etc. This $\triangle T_{A}T_{B}T_{C}$ is also known as theextouch triangle of $\triangle ABC$ . Thecircumcircle of the extouch $\triangle T_{A}T_{B}T_{C}$ is called theMandart circle(cf.Mandart inellipse).

The three line segments ${\overline {AT_{A}}}$ , ${\overline {BT_{B}}}$ and ${\overline {CT_{C}}}$ are called thesplitters of the triangle; they each bisect the perimeter of the triangle,^{[citation needed]}

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

The splitters intersect in a single point, the triangle'sNagel point $N_{a}$ (ortriangle centerX₈).

Trilinear coordinates for the vertices of the extouch triangle are given by^{[citation needed]}

{\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}}

Trilinear coordinates for the Nagel point are given by^{[citation needed]}

\csc ^{2}{\tfrac {A}{2}}:\csc ^{2}{\tfrac {B}{2}}:\csc ^{2}{\tfrac {C}{2}},

or, equivalently, by theLaw of Sines,

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

The Nagel point is theisotomic conjugate of the Gergonne point.^{[citation needed]}

Related constructions

Nine-point circle and Feuerbach point

Main article:Nine-point circle

The nine-point circle is tangent to the incircle and excircles

Ingeometry, thenine-point circle is acircle that can be constructed for any giventriangle. It is so named because it passes through nine significantconcyclic points defined from the triangle. These ninepoints are:^[28]^[29]

Themidpoint of each side of the triangle
Thefoot of eachaltitude
The midpoint of theline segment from eachvertex of the triangle to theorthocenter (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 externallytangent to that triangle's three excircles and internally tangent to its incircle; this result is known asFeuerbach's theorem. He proved that:^[30]

... 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)

Thetriangle center at which the incircle and the nine-point circle touch is called theFeuerbach point.

The incircle may be described as thepedal circle of the incenter. The locus of points whose pedal circles are tangent to the nine-point circle is known as theMcCay cubic.

Incentral and excentral triangles

The points of intersection of the interior angle bisectors of $\triangle ABC$ with the segments $B C {\displaystyle BC}$ , $C A {\displaystyle CA}$ , and $A B {\displaystyle AB}$ are the vertices of theincentral triangle. Trilinear coordinates for the vertices of the incentral triangle $\triangle A'B'C'$ are given by^{[citation needed]}

{\begin{array}{ccccccc}A'&=&0&:&1&:&1\\[2pt]B'&=&1&:&0&:&1\\[2pt]C'&=&1&:&1&:&0\end{array}}

Theexcentral 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 attop of page). Trilinear coordinates for the vertices of the excentral triangle $\triangle A'B'C'$ are given by^{[citation needed]}

{\begin{array}{ccrcrcr}A'&=&-1&:&1&:&1\\[2pt]B'&=&1&:&-1&:&1\\[2pt]C'&=&1&:&1&:&-1\end{array}}

Equations for four circles

Let $x : y : z {\displaystyle x:y:z}$ be a variable point intrilinear coordinates, and let $u=\cos ^{2}\left(A/2\right)$ , $v=\cos ^{2}\left(B/2\right)$ , $w=\cos ^{2}\left(C/2\right)$ . The four circles described above are given equivalently by either of the two given equations:^[31]^{: 210–215}

Incircle: ${\begin{aligned}u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy&=0\\[4pt]{\textstyle \pm {\sqrt {x}}\cos {\tfrac {A}{2}}\pm {\sqrt {y{\vphantom {t}}}}\cos {\tfrac {B}{2}}\pm {\sqrt {z}}\cos {\tfrac {C}{2}}}&=0\end{aligned}}$
$A {\displaystyle A}$ -excircle: ${\begin{aligned}u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz+2wuzx+2uvxy&=0\\[4pt]{\textstyle \pm {\sqrt {-x}}\cos {\tfrac {A}{2}}\pm {\sqrt {y{\vphantom {t}}}}\cos {\tfrac {B}{2}}\pm {\sqrt {z}}\cos {\tfrac {C}{2}}}&=0\end{aligned}}$
$B {\displaystyle B}$ -excircle: ${\begin{aligned}u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz-2wuzx+2uvxy&=0\\[4pt]{\textstyle \pm {\sqrt {x}}\cos {\tfrac {A}{2}}\pm {\sqrt {-y{\vphantom {t}}}}\cos {\tfrac {B}{2}}\pm {\sqrt {z}}\cos {\tfrac {C}{2}}}&=0\end{aligned}}$
$C {\displaystyle C}$ -excircle: ${\begin{aligned}u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz+2wuzx-2uvxy&=0\\[4pt]{\textstyle \pm {\sqrt {x}}\cos {\tfrac {A}{2}}\pm {\sqrt {y{\vphantom {t}}}}\cos {\tfrac {B}{2}}\pm {\sqrt {-z}}\cos {\tfrac {C}{2}}}&=0\end{aligned}}$

Euler's theorem

Euler's theorem states that in a triangle:

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

where $R {\displaystyle R}$ and $r {\displaystyle r}$ are the circumradius and inradius respectively, and $d {\displaystyle d}$ is the distance between thecircumcenter and the incenter.

For excircles the equation is similar:

\left(R+r_{\text{ex}}\right)^{2}=d_{\text{ex}}^{2}+r_{\text{ex}}^{2},

where $r_{\text{ex}}$ is the radius of one of the excircles, and $d_{\text{ex}}$ is the distance between the circumcenter and that excircle's center.^[32]^[33]^[34]

Generalization to other polygons

Some (but not all)quadrilaterals have an incircle. These are calledtangential quadrilaterals. Among their many properties perhaps the most important is that their two pairs of opposite sides have equal sums. This is called thePitot theorem.^[35]

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 atangential polygon.

Generalization to topological triangles

If you considertopological triangles, you can also define a notion of inscribed circle that fits inside. It is no longer described as tangent to all sides since your topological triangle might not be differentiable everywhere. Rather it is defined as a circle whose center has the same minimal distance to each side. Its a proven fact that an inscribed circle always exists in any topological triangle.^[36]

See also

Circumcenter – Circle that passes through the vertices of a trianglePages displaying short descriptions of redirect targets
Circumcircle – Circle that passes through the vertices of a triangle
Circumconic and inconic – Conic section that passes through the vertices of a triangle or is tangent to its sides
Circumgon – Geometric figure which circumscribes a circle
Ex-tangential quadrilateral – Convex 4-sided polygon whose sidelines are all tangent to an outside circle
Harcourt's theorem – Area of a triangle from its sides and vertex distances to any line tangent to its incircle
Incenter–excenter lemma – Theorem about inscribed and circumscribed circles
Inscribed sphere – Sphere tangent to every face of a polyhedron
Power of a point – Relative distance of a point from a circle
Steiner inellipse – Unique ellipse tangent to all 3 midpoints of a given triangle's sides
Tangential quadrilateral – Polygon whose four sides all touch a circle
Triangle conic

Notes

^Kay (1969, p. 140)
^^a ^bAltshiller-Court (1925, p. 74)
^^a ^b ^c ^d ^eAltshiller-Court (1925, p. 73)
^Kay (1969, p. 117)
^^a ^bJohnson 1929, p. 182.
^^a ^bEncyclopedia of Triangle Centers Archived 2012-04-19 at theWayback Machine, accessed 2014-10-28.
^Kay (1969, p. 201)
^Chu, Thomas,The Pentagon, Spring 2005, p. 45, problem 584.
^Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity",Mathematical Gazette,96:161–165,doi:10.1017/S0025557200004277,S2CID 124176398.
^Altshiller-Court, Nathan (1980),College Geometry, Dover Publications. #84, p. 121.
^Mathematical Gazette, July 2003, 323-324.
^Kay (1969, p. 203)
^Johnson 1929, p. 189, #298(d).
^^a ^b ^c ^dBell, Amy.""Hansen's right triangle theorem, its converse and a generalization",Forum Geometricorum 6, 2006, 335–342"(PDF). Archived fromthe original(PDF) on 2021-08-31. Retrieved2012-05-05.
^Kodokostas, Dimitrios, "Triangle Equalizers",Mathematics Magazine 83, April 2010, pp. 141-146.
^Posamentier, Alfred S., and Lehmann, Ingmar.The Secrets of Triangles, Prometheus Books, 2012.
^^a ^b ^cFranzsen, William N. (2011)."The distance from the incenter to the Euler line"(PDF).Forum Geometricorum.11:231–236.MR 2877263. Archived fromthe original(PDF) on 2020-12-05. Retrieved2012-05-09..
^Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
^Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials",American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
^Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/ContactTriangle.html
^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 Archived 2016-03-04 at theWayback Machine
^Dekov, Deko (2009)."Computer-generated Mathematics : The Gergonne Point"(PDF).Journal of Computer-generated Euclidean Geometry.1:1–14. Archived fromthe original(PDF) on 2010-11-05.
^^a ^bAltshiller-Court (1925, p. 79)
^Kay (1969, p. 202)
^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.)
^"Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle",Forum Geometricorum 2, 2002: pp. 175-182"(PDF). Archived fromthe original(PDF) on 2023-06-25. Retrieved2012-05-02.
^"Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers",Forum Geometricorum 3, 2003, 187-195"(PDF). Archived fromthe original(PDF) on 2022-10-06. Retrieved2012-05-03.
^Altshiller-Court (1925, pp. 103–110)
^Kay (1969, pp. 18, 245)
^Feuerbach, Karl Wilhelm; Buzengeiger, Carl Heribert Ignatz (1822),Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung (in German) (Monograph ed.), Nürnberg: Wiessner.
^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
^Nelson, Roger, "Euler's triangle inequality via proof without words",Mathematics Magazine 81(1), February 2008, 58-61.
^Johnson 1929, p. 187.
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^Josefsson (2011, See in particular pp. 65–66.)
^Al-Ajrawi, Ezzaddin (20 October 2024)."Inscribing Spheres in Topologically Embedded Simplices".The American Mathematical Monthly.131 (9):806–813.doi:10.1080/00029890.2024.2380233.ISSN 0002-9890.

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
Johnson, Roger A. (1929),"X. Inscribed and Escribed Circles",Modern Geometry, Houghton Mifflin, pp. 182–194
Josefsson, Martin (2011),"More characterizations of tangential quadrilaterals"(PDF),Forum Geometricorum,11:65–82,MR 2877281, archived fromthe original(PDF) on 2016-03-04, retrieved2023-03-14
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.

External links

Interactive

Triangle incenter;Triangle incircle;Incircle of a regular polygon (with interactive animations)
Constructing a triangle's incenter / incircle with compass and straightedge An interactive animated demonstration
Equal Incircles Theorem atcut-the-knot
Five Incircles Theorem atcut-the-knot
Pairs of Incircles in a Quadrilateral atcut-the-knot
An interactive Java applet for the incenter

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