Incircle redirects here. For incircles of non-triangle polygons, see Tangential
quadrilateral or Tangential polygon. In geometry, the incircle or inscribed
circle of a triangle is the largest circle contained in the triangle; it
touches the three sides. The center of the incircle is 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 angle bisectors. The center of an excircle is
the intersection of the internal bisector of one angle 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. Polygons with more than three sides do
not all have an incircle tangent to all sides; those that do are called
tangential polygons. See also Tangent lines to circles.
Relation to area of the triangle The radii of the incircles and excircles
are closely related to the area of the triangle.
= Incircle = Suppose has an incircle with radius r
and center I. Let a be the length of BC, b the length of AC, and c the length of
AB. Now, the incircle is tangent to AB at some point C′, and so is right. Thus
the radius C'I is an altitude of . Therefore has base length c and height
r, and so has area . Similarly, has area and has area . Since these three
triangles decompose , we see that and
where is the area of and is its semiperimeter.
For an alternative formula, consider . This is a right-angled triangle with one
side equal to r and the other side equal to . The same is true for . The large
triangle is composed of 6 such triangles and the total area is:
= Excircles = The radii in the excircles are called
the exradii. Let the excircle at side AB touch at side AC extended at G, and let
this excircle's radius be and its center be . Then is an altitude of , so
has area . By a similar argument, has area and has area . Thus
So, by symmetry, 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 From these formulas one can see that the
excircles are always larger than the incircle and that the largest excircle
is the one tangent to the longest side and the smallest excircle is tangent to
the shortest side. Further, combining these formulas yields:
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.
Related constructions = Nine-point circle and Feuerbach point
The circle tangent to all three of the excircles as well as the incircle is
known as the nine-point circle. The point where the nine-point circle
touches the incircle is known as the Feuerbach point.
= Gergonne triangle and point = The Gergonne triangle is defined by the
3 touchpoints of the incircle on the 3 sides. The touchpoint opposite A is
denoted TA, etc. This Gergonne triangle TATBTC is also
known as the contact triangle or intouch triangle of ABC.
The three lines ATA, BTB and CTC intersect in a single point called
Gergonne point, denoted as Ge - X(7). The Gergonne point lies in the open
orthocentroidal disk punctured at its own center, and could be any point
therein. Interestingly, the Gergonne point of a
triangle is the symmedian point of the Gergonne triangle. For a full set of
properties of the Gergonne point see. 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,
= Nagel triangle and point = The Nagel triangle of ABC is denoted by
the vertices XA, XB and XC that are the three points where the excircles touch
the reference triangle ABC and where XA is opposite of A, etc. This triangle
XAXBXC is also known as the extouch triangle of ABC. The circumcircle of the
extouch triangle XAXBXC is called the Mandart circle. The three lines AXA, BXB
and CXC are called the splitters of the triangle; they each bisect the perimeter
of the triangle, and they intersect in a single point, the triangle's Nagel point
Na - X(8). Trilinear coordinates for the vertices
of the extouch triangle are given by Trilinear coordinates for the Nagel
point are given by or, equivalently, by the Law of Sines,
It is the isotomic conjugate of the Gergonne point.
= Incentral and excentral triangles = The points of intersection of the
interior angle bisectors of ABC with the segments BC, CA, AB are the vertices of
the incentral triangle. Trilinear coordinates for the vertices
of the incentral triangle are given by Trilinear coordinates for the vertices
of the excentral triangle are given by Equations for four circles
Let x : y : z be a variable point in trilinear coordinates, and let u =
cos2(A/2), v = cos2(B/2), w = cos2(C/2). The four circles described above are
given equivalently by either of the two given equations:
Incircle: A-excircle:
B-excircle: C-excircle:
Euler's theorem Euler's theorem states that in a
triangle: where R and rin are the circumradius and
inradius respectively, and d is the distance between the circumcenter and
the incenter. For excircles the equation is similar:
where rex is the radius of one of the excircles, and d is the distance between
the circumcenter and this excircle's center.
Other incircle properties Suppose the tangency points of the
incircle divide the sides into lengths of x and y, y and z, and z and x. Then
the incircle has the radius and the area of the triangle is
If the altitudes from sides of lengths a, b, and c are ha, hb, and hc then the
inradius r is one-third of the harmonic mean of these altitudes, i.e.
The product of the incircle radius r and the circumcircle radius R of a triangle
with sides a, b, and c is 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. 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 and
The distance from any vertex to the incircle tangency on either adjacent
side is half the sum of the vertex's adjacent sides minus half the opposite
side. Thus for example for vertex B and adjacent tangencies TA and TC,
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
and the distance from the incenter to the center N of the nine point circle is
The incenter lies in the medial triangle.
Other excircle properties The circular hull of the excircles is
internally tangent to each of the excircles, and thus is an Apollonius
circle. The radius of this Apollonius circle is 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 ra, rb, rc:
The circle through the centers of the three excircles has radius 2R.
If H is the orthocenter of triangle ABC, then
Generalization to other polygons Some quadrilaterals have an incircle.
These are called tangential quadrilaterals. Among their many
properties perhaps the most important is that their two pairs of opposite sides
have equal sums. This is called the Pitot theorem.
More generally, a polygon with any number of sides that has an inscribed
circle—one that is tangent to each side—is called a tangential polygon.
See also Altitude
Circumscribed circle Ex-tangential quadrilateral
Harcourt's theorem Inconic
Inscribed sphere Power of a point
Steiner inellipse Tangential quadrilateral
Triangle center Notes
References Altshiller-Court, Nathan, College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle,
New York: Barnes & Noble, LCCN 52013504 Kay, David C., College Geometry, New
York: Holt, Rinehart and Winston, LCCN 69012075
Kimberling, Clark. "Triangle Centers and Central Triangles". Congressus
Numerantium: i–xxv,1–295. Kiss, Sándor. "The Orthic-of-Intouch and
Intouch-of-Orthic Triangles". Forum Geometricorum: 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