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 has an incircle with radius and center .
Let be the length of , the length of , and the length of .
Also let , , and be the touchpoints where the incircle touches , , and .
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 meet.
The distance from vertex to the incenter is:
:
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[Encyclopedia of Triangle Centers](_blank)
, accessed 2014-10-28.
:
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
:
where , , and are the lengths of the sides of the triangle, or equivalently (using the law of sines) by
:
where , , and 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 , , and , and the sides opposite these vertices have corresponding lengths , , and , then the incenter is at
:
Radius
The inradius of the incircle in a triangle with sides of length '', '', is given by
: where
See Heron's formula.
Distances to the vertices
Denoting the incenter of as , the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation
:
Additionally,
:
where and 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.]
:
Other properties
Suppose the tangency points of the incircle divide the sides into lengths of and , and , and '''' and . Then the incircle has the radius
:
and the area of the triangle is
:
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 '', '', and are , , and '''', then the inradius '''' is one-third of the harmonic mean of these altitudes; that is,
:
The product of the incircle radius '''' 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 of a triangle with sides '', '', and is[Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover, 2007 (orig. 1929).]
:
Some relations among the sides, incircle radius, and circumcircle radius are:[
:
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 as , we have
:
and
:
The incircle radius is no greater than one-ninth the sum of the altitudes.
The squared distance from the incenter to the circumcenter is given by][.]
:,
and the distance from the incenter to the center of the nine point circle is[
:
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
,
with equality holding only for equilateral triangles.
Suppose
has an incircle with radius and center . Let be the length of , the length of , and the length of ''.'' Now, the incircle is tangent to '''' at some point , and so
is right. Thus, the radius 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
.
Therefore,
has base length '''' and height , and so has area
.
Similarly,
has area
and
has area
.
Since these three triangles decompose
, we see that the area
is:
: and
where is the area of and 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 . This is a right-angled triangle with one side equal to '''' and the other side equal to . The same is true for . The large triangle is composed of six such triangles and the total area is:
:
Gergonne triangle and point
The Gergonne triangle (of '''') is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite is denoted '''', etc.
This Gergonne triangle, '''', is also known as the contact triangle or intouch triangle of ''''. Its area is
:
where , , and are the area, radius of the incircle, and semiperimeter of the original triangle, and , , and 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 '''', '''' and '''' intersect in a single point called the Gergonne point, denoted as '''' (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
*
*
*
Trilinear coordinates for the Gergonne point are given by
:
or, equivalently, by the Law of Sines,
:
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 , 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.[
]
Trilinear coordinates of excenters
While the incenter of '''' has trilinear coordinates , the excenters have trilinears , , and .
Exradii
The radii of the excircles are called the exradii.
The exradius of the excircle opposite (so touching , centered at ) is
: where
See Heron's formula.
Derivation of exradii formula
Let the excircle at side touch at side extended at , and let this excircle's
radius be and its center be .
Then
is an altitude of
,
so
has area
.
By a similar argument,
has area
and
has area
.
Thus the area
of triangle
is
: .
So, by symmetry, denoting as the radius of the incircle,
: .
By the Law of Cosines, we have
:
Combining this with the identity , we have
:
But , and so
:
which is Heron's formula.
Combining this with , we have
:
Similarly, gives
:
and
:
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:
:
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 where is the incircle radius and is the semiperimeter of the triangle.
The following relations hold among the inradius '''', the circumradius , the semiperimeter '''', and the excircle radii '''', '''', '''':[Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342.]
/ref>
:
The circle through the centers of the three excircles has radius .[
If '''' is the orthocenter of '''', then][
:
]
Nagel triangle and Nagel point
The Nagel triangle or extouch triangle of '''' is denoted by the vertices , , and that are the three points where the excircles touch the reference '''' and where '''' is opposite of '''', etc. This '''' is also known as the extouch triangle of ''''. 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 '''' is called the Mandart circle.
The three lines , and are called the splitters of the triangle; they each bisect the perimeter of the triangle,
:
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 ...
(or triangle center ''X''8).
Trilinear coordinates for the vertices of the extouch triangle are given by
*
*
*
Trilinear coordinates for the Nagel point are given by
:
or, equivalently, by the Law of Sines,
:
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 '''' with the segments '', ,'' and '''' are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle are given by
*
*
*
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
*
*
*
Equations for four circles
Let '''' be a variable point in trilinear coordinates, and let '''', '''', ''''. 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:
*:
*''-''excircle:
*:
* ''-''excircle:
*:
*''-''excircle:
*:
Euler's theorem
Euler's theorem states that in a triangle:
:
where '''' and '''' are the circumradius and inradius respectively, and '''' 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:
:
where '''' is the radius of one of the excircles, and '''' 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
nbsp; &nbs
nbsp;&nbs
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