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In geometry, the incircle or inscribed circle of a triangle is the largest
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
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. 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 Internal may refer to: *Internality as a concept in behavioural economics *Neijia, internal styles of Chinese martial arts *Neigong or "internal skills", a type of exercise in meditation associated with Daoism *''Internal (album)'' by Safia, 2016 ...
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
s. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . 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. but not all polygons do; those that do are
tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...
s. See also tangent lines to circles.


Incircle and incenter

Suppose \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 In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of \angle ABC, \angle BCA, \text \angle BAC meet. The distance from vertex A to the incenter I is: : d(A, I) = c \frac = b \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.
:\ 1 : 1 : 1.


Barycentric coordinates

The
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
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 where a, b, and c are the lengths of the sides of the triangle, or equivalently (using the law of sines) by :\sin(A):\sin(B):\sin(C) where A, B, and C are the angles at the three vertices.


Cartesian coordinates

The
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
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 : \left(\frac, \frac\right) = \frac.


Radius

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


Distances 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 :\frac + \frac + \frac = 1. Additionally, :IA \cdot IB \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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
.


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\left(A, T_B\right) = d\left(A, T_C\right) = \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 = \sqrt and the area of the triangle is :\Delta = \sqrt. If the
altitudes Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
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 = \frac. The product of the incircle radius ''r '' and the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
radius R of a triangle with sides a'', b'', and c isJohnson, Roger A., ''Advanced Euclidean Geometry'', Dover, 2007 (orig. 1929). :rR = \frac. Some relations among the sides, incircle radius, and circumcircle radius are: :\begin ab + bc + ca &= s^2 + (4R + r)r, \\ a^2 + b^2 + c^2 &= 2s^2 - 2(4R + r)r. \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 \triangle ABC as I , we have :\frac + \frac + \frac = 1 and :IA \cdot IB \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 = \frac(R - 2r) < \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 Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
of the triangle. The ratio of the area of the incircle to the area of the triangle is less than or equal to \tfrac, with equality holding only for equilateral triangles. Suppose \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 \angle AT_CI is right. Thus, the radius T_CI is an
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
of \triangle IAB. Therefore, \triangle IAB has base length ''c'' and height r, and so has area \tfraccr. Similarly, \triangle IAC has area \tfracbr and \triangle IBC has area \tfracar. Since these three triangles decompose \triangle ABC, we see that the area \Delta \text \triangle ABC is: :\Delta = \frac (a + b + c)r = sr, and r = \frac, where \Delta is the area of \triangle ABC and s = \tfrac(a + b + c) is its
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate ...
. For an alternative formula, consider \triangle IT_CA. This is a right-angled triangle with one side equal to ''r'' and the other side equal to r \cot\left(\frac\right). The same is true for \triangle IB'A . The large triangle is composed of six such triangles and the total area is: :\Delta = r^2 \left(\cot\left(\frac\right) + \cot\left(\frac\right) + \cot\left(\frac\right)\right).


Gergonne triangle and point

The Gergonne triangle (of ''\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, ''\triangle T_AT_BT_C'', is also known as the contact triangle or intouch triangle of ''\triangle ABC''. Its area is :K_T = K\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 In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. Coordinates The vertices of the extouch triangle are given in trilinear coordinates by: :\begin T_A &= 0 ...
. 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 * \text\, T_A = 0 : \sec^2 \left(\frac\right) : \sec^2\left(\frac\right) * \text\, T_B = \sec^2 \left(\frac\right) : 0 : \sec^2\left(\frac\right) * \text\, T_C = \sec^2 \left(\frac\right) : \sec^2\left(\frac\right) : 0. Trilinear coordinates for the Gergonne point are given by :\sec^2\left(\frac\right) : \sec^2 \left(\frac\right) : \sec^2\left(\frac\right), or, equivalently, by the Law of Sines, :\frac : \frac : \frac.


Excircles and excenters

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 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 ''\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 = \frac = \sqrt, where s = \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 \triangle ACJ_c, so \triangle ACJ_c has area \tfracbr_c. By a similar argument, \triangle BCJ_c has area \tfracar_c and \triangle ABJ_c has area \tfraccr_c. Thus the area \Delta of triangle \triangle ABC is : \Delta = \frac(a + b - c)r_c = (s - c)r_c. So, by symmetry, denoting r as the radius of the incircle, : \Delta = sr = (s - a)r_a = (s - b)r_b = (s - c)r_c. By the Law of Cosines, we have : \cos(A) = \frac Combining this with the identity \sin^2 A + \cos^2 A = 1, we have : \sin(A) = \frac But \Delta = \tfracbc \sin(A), and so :\begin \Delta &= \frac \sqrt \\ &= \frac \sqrt \\ & = \sqrt, \end which is Heron's formula. Combining this with sr = \Delta, we have :r^2 = \frac = \frac. Similarly, (s - a)r_a = \Delta gives :r_a^2 = \frac and :r_a = \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: :\Delta = \sqrt.


Other excircle properties

The circular
hull Hull may refer to: Structures * Chassis, of an armored fighting vehicle * Fuselage, of an aircraft * Hull (botany), the outer covering of seeds * Hull (watercraft), the body or frame of a ship * Submarine hull Mathematics * Affine hull, in affi ...
of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle. The radius of this Apollonius circle is \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> :\begin 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 The circle through the centers of the three excircles has radius 2R. If ''H'' is the orthocenter of ''\triangle ABC'', then :\begin r_a + r_b + r_c + r &= AH + BH + CH + 2R, \\ r_a^2 + r_b^2 + r_c^2 + r^2 &= AH^2 + BH^2 + CH^2 + (2R)^2. \end


Nagel triangle and Nagel point

The Nagel triangle or extouch 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'', etc. This ''\triangle T_AT_BT_C'' is also known as the extouch triangle of ''\triangle ABC''. The
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
of the extouch ''\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 = \frac\left( AB + BC + AC \right). The splitters intersect in a single point, the triangle's
Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurr ...
N_a (or triangle center ''X''8). Trilinear coordinates for the vertices of the extouch triangle are given by * \text \, T_A = 0 : \csc^2\left(\frac\right) : \csc^2\left(\frac\right) * \text \, T_B = \csc^2\left(\frac\right) : 0 : \csc^2\left(\frac\right) *\text \, T_C = \csc^2\left(\frac\right) : \csc^2\left(\frac\right) : 0. Trilinear coordinates for the Nagel point are given by :\csc^2\left(\frac\right) : \csc^2 \left(\frac\right) : \csc^2\left(\frac\right), or, equivalently, by the Law of Sines, :\frac : \frac : \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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
that can be constructed for any given triangle. It is so named because it passes through nine significant
concyclic points In geometry, a set of points are said to be concyclic (or cocyclic) if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line ...
defined from the triangle. These nine
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points ...
are: * The
midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...
of each side of the triangle * The
foot The foot ( : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg made ...
of each
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
* 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 ''\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 * \ \left( \text \, A\right) = 0 : 1 : 1 * \ \left( \text \, B\right) = 1 : 0 : 1 *\ \left( \text \, C\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 * (\text \, A) = -1 : 1 : 1 * (\text \, B) = 1 : -1 : 1 *(\text \, C) = 1 : 1 : -1.


Equations for four circles

Let ''x:y:z'' be a variable point in trilinear 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: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: *:\begin u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz - 2wuzx - 2uvxy &= 0 \\ \pm\sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) &= 0 \end *''A-''excircle: *:\begin u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz + 2wuzx + 2uvxy &= 0 \\ \pm\sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) &= 0 \end * ''B-''excircle: *:\begin u^2 x^2 + v^2 y^2 + w^2 z^2 + 2vwyz - 2wuzx + 2uvxy &= 0 \\ \pm\sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) &= 0 \end *''C-''excircle: *:\begin u^2 x^2 + v^2 y^2 + w^2 z^2 + 2vwyz + 2wuzx - 2uvxy &= 0 \\ \pm\sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) \pm \sqrt\cos\left(\frac\right) &= 0 \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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
and the incenter. For excircles the equation is similar: :\left(R + r_\text\right)^2 = d_\text^2 + r_\text^2, where ''r_\text'' is the radius of one of the excircles, and ''d_\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 In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...
.


See also

* * * * * * * * * *
Triangle conic In triangle geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steine ...
*


Notes


References

* * * *


External links


Derivation of formula for radius of incircle of a triangle
*


Interactive



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nbsp;  With interactive animations

An interactive animated demonstration
Equal Incircles Theorem
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Five Incircles Theorem
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Pairs of Incircles in a Quadrilateral
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

An interactive Java applet for the incenter
{{DEFAULTSORT:Incircle And Excircles Of A Triangle Circles defined for a triangle