Incircle and excircles

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.^[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] but not all polygons do; those that do are tangential polygons. See also tangent lines to circles.

Incircle and incenter

Suppose ${\displaystyle \triangle ABC}$ has an incircle with radius ${\displaystyle r}$ and center ${\displaystyle I}$. Let ${\displaystyle a}$ be the length of ${\displaystyle {\overline {BC}}}$, ${\displaystyle b}$ the length of ${\displaystyle {\overline {AC}}}$, and ${\displaystyle c}$ the length of ${\displaystyle {\overline {AB}}}$. Also let ${\displaystyle T_{A}}$, ${\displaystyle T_{B}}$, and ${\displaystyle T_{C}}$ be the touchpoints where the incircle touches ${\displaystyle {\overline {BC}}}$, ${\displaystyle {\overline {AC}}}$, and ${\displaystyle {\overline {AB}}}$.

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 ${\displaystyle A}$ to the incenter ${\displaystyle I}$ is:^{[citation needed]}

Trilinear coordinates

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

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

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

where ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ 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 ${\displaystyle (x_{a},y_{a})}$, ${\displaystyle (x_{b},y_{b})}$, 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^{[citation needed]}

Radius

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

where ${\displaystyle s={\tfrac {1}{2}}(a+b+c)}$ is the semiperimeter.

The tangency points of the incircle divide the sides into segments of lengths ${\displaystyle s-a,}$ ${\displaystyle s-b,}$ and ${\displaystyle s-c.}$^[8]

See Heron's formula.

Distances to the vertices

Denoting the incenter of ${\displaystyle \triangle ABC}$ as ${\displaystyle I}$, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation^[9]

Additionally,^[10]

where ${\displaystyle R}$ and ${\displaystyle r}$ 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.^[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]

Other properties

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

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

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

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]

Denoting the center of the incircle of ${\displaystyle \triangle ABC}$ as ${\displaystyle I}$, we have^[16]

and^{[17]: 121, #84}

The incircle radius is no greater than one-ninth the sum of the altitudes.^[18]: 289

The squared distance from the incenter ${\displaystyle I}$ to the circumcenter ${\displaystyle O}$ is given by^[19]: 232

and the distance from the incenter to the center ${\displaystyle N}$ of the nine point circle is^[19]: 232

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).^{[19]: 233, Lemma 1}

Relation to area of the triangle

The radius of the incircle is related to the area of the triangle.^[20] The ratio of the area of the incircle to the area of the triangle is less than or equal to ${\displaystyle \pi {\big /}3{\sqrt {3}}}$, with equality holding only for equilateral triangles.^[21]

Suppose ${\displaystyle \triangle ABC}$ has an incircle with radius ${\displaystyle r}$ and center ${\displaystyle I}$. Let ${\displaystyle a}$ be the length of ${\displaystyle {\overline {BC}}}$, ${\displaystyle b}$ the length of ${\displaystyle {\overline {AC}}}$, and ${\displaystyle c}$ the length of ${\displaystyle {\overline {AB}}}$. Now, the incircle is tangent to ${\displaystyle {\overline {AB}}}$ at some point ${\displaystyle T_{C}}$, and so ${\displaystyle \angle AT_{C}I}$ is right. Thus, the radius ${\displaystyle T_{C}I}$ is an altitude of ${\displaystyle \triangle IAB}$. Therefore, ${\displaystyle \triangle IAB}$ has base length ${\displaystyle c}$ and height ${\displaystyle r}$, 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:

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

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

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

Gergonne triangle and point

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

This Gergonne triangle, ${\displaystyle \triangle T_{A}T_{B}T_{C}}$, is also known as the contact triangle or intouch triangle of ${\displaystyle \triangle ABC}$. Its area is

where ${\displaystyle K}$, ${\displaystyle r}$, and ${\displaystyle s}$ are the area, radius of the incircle, and semiperimeter of the original triangle, and ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are the side lengths of the original triangle. This is the same area as that of the extouch triangle.^[22]

The three lines ${\displaystyle AT_{A}}$, ${\displaystyle BT_{B}}$ and ${\displaystyle CT_{C}}$ intersect in a single point called the Gergonne point, denoted as ${\displaystyle G_{e}}$ (or triangle center X₇). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.^[23]

The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.^[24]

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

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

or, equivalently, by the Law of Sines,

Excircles and excenters

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 an excircle is the intersection of the internal bisector of one angle (at vertex ${\displaystyle A}$, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex ${\displaystyle A}$, or the excenter of ${\displaystyle A}$.^[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 an orthocentric system.^[5]

Trilinear coordinates of excenters

While the incenter of ${\displaystyle \triangle ABC}$ has trilinear coordinates ${\displaystyle 1:1:1}$, the excenters have trilinears ^{[citation needed]}

Exradii

The radii of the excircles are called the exradii.

The exradius of the excircle opposite ${\displaystyle A}$ (so touching ${\displaystyle BC}$, centered at ${\displaystyle J_{A}}$) is^[25][26]

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

See Heron's formula.

Derivation of exradii formula

Source:^[25]

Let the excircle at side ${\displaystyle AB}$ touch at side ${\displaystyle AC}$ extended at ${\displaystyle G}$, 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

.

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

.

By the Law of Cosines, we have

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

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

which is Heron's formula.

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

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

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:^[27]

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.^[28] The radius of this Apollonius circle is ${\displaystyle {\tfrac {r^{2}+s^{2}}{4r}}}$ where ${\displaystyle r}$ is the incircle radius and ${\displaystyle s}$ is the semiperimeter of the triangle.^[29]

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

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

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

Nagel triangle and Nagel point

The Nagel triangle or extouch triangle of ${\displaystyle \triangle ABC}$ is denoted by the vertices ${\displaystyle T_{A}}$, ${\displaystyle T_{B}}$, and ${\displaystyle T_{C}}$ that are the three points where the excircles touch the reference ${\displaystyle \triangle ABC}$ and where ${\displaystyle T_{A}}$ is opposite of ${\displaystyle A}$, etc. This ${\displaystyle \triangle T_{A}T_{B}T_{C}}$ is also known as the extouch triangle of ${\displaystyle \triangle ABC}$. The circumcircle of the extouch ${\displaystyle \triangle T_{A}T_{B}T_{C}}$ is called the Mandart circle.^{[citation needed]}

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

The splitters intersect in a single point, the triangle's Nagel point ${\displaystyle N_{a}}$ (or triangle center X₈).

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

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

or, equivalently, by the Law of Sines,

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

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:^[30][31]

• 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:^[32]

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

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 ${\displaystyle \triangle A'B'C'}$ are given by^{[citation needed]}

Equations for four circles

Let ${\displaystyle x:y:z}$ be a variable point in trilinear coordinates, and let ${\displaystyle u=\cos ^{2}\left(A/2\right)}$, ${\displaystyle v=\cos ^{2}\left(B/2\right)}$, ${\displaystyle w=\cos ^{2}\left(C/2\right)}$. The four circles described above are given equivalently by either of the two given equations:^{[33]: 210–215}

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

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

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

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

  

Euler's theorem

Euler's theorem states that in a triangle:

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

For excircles the equation is similar:

where ${\displaystyle r_{\text{ex}}}$ is the radius of one of the excircles, and ${\displaystyle d_{\text{ex}}}$ is the distance between the circumcenter and that excircle's center.^[34][35][36]

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.^[37]

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.^{[citation needed]}

See also

• Circumgon – Geometric figure which circumscribes a circle
• Circumcircle – Circle that passes through the vertices of a triangle
• 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
• Circumconic and inconic – Conic section that passes through the vertices of a triangle or is tangent to its sides
• 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
• Trillium theorem – A statement about properties of inscribed and circumscribed circles

Notes

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
• 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

• Derivation of formula for radius of incircle of a triangle
• Weisstein, Eric W. "Incircle". MathWorld.

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 at cut-the-knot
• Five Incircles Theorem at cut-the-knot
• Pairs of Incircles in a Quadrilateral at cut-the-knot
• An interactive Java applet for the incenter