Incircle and excircles of a triangle

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the incircle or inscribed circle of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

is the largest circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

contained in the triangle; it touches (is tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to) 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 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. 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

Orthocentric system

In geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three....

.

See also Tangent lines to circles

Tangent lines to circles

In Euclidean plane geometry, tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs...

Relation to area of the triangle

The radii of the in- and excircles are closely related to the area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

of the triangle. Let K be the triangle's area and let a, b and c, be the lengths of its sides. By Heron's formula, the area of the triangle is

where

is the semiperimeter and P = 2s is the perimeter.

The radius of the incircle (also known as the inradius, r ) is

Thus, the area K of a triangle may be found by multiplying the inradius by the semiperimeter:

The radii in the excircles are called the exradii. The excircle at side a has radius

Similarly the radii of the excircles at sides b and c are respectively

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 with Heron's area formula yields the result that

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

Nine-point circle

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 points defined from the triangle...

. The point where the nine-point circle touches the incircle is known as the Feuerbach point.

Gergonne triangle and point

The Gergonne triangle of ABC is denoted by the vertices T_A, T_B and T_C that are the three points where the incircle touches the reference triangle ABC and where T_A is opposite of A, etc. This triangle T_AT_BT_C is also known as the contact triangle or intouch triangle of ABC. The incircle of ABC is the circumcircle of T_AT_BT_C. The three lines AT_A, BT_B and CT_C intersect in a single point, the triangle's Gergonne point Ge - X(7)

Triangle center

In geometry a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of...

. Interestingly, the Gergonne point of a triangle is the symmedian point of its Gergonne triangle. For a full set of properties of the Gergonne point see.

The touchpoints of the excircle with segments BC,CA,AB are the vertices of the extouch triangle. The points of intersection of the interior angle bisectors of ABC with the segments BC,CA,AB are the vertices of the incentral triangle.

Nagel triangle and point

The Nagel triangle of ABC is denoted by the vertices X_A, X_B and X_C that are the three points where the excircles touches the reference triangle ABC and where X_A is opposite of A, etc. This triangle X_AX_BX_C is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle X_AX_BX_C is called the Mandart circle. The three lines AX_A, BX_B and CX_C intersect in a single point, the triangle's Nagel point

Nagel point

In geometry, the Nagel point is a point associated with any triangle. Given a triangle ABC, let TA, TB, and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively...

Na - X(8)

Triangle center

.

Trilinear coordinates

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...

for the vertices of the intouch triangle are given by

Trilinear coordinates

for the vertices of the extouch triangle are given by

Trilinear coordinates

for the vertices of the incentral triangle are given by

Trilinear coordinates

for the vertices of the excentral triangle are given by

Trilinear coordinates

for the Gergonne point are given by

or, equivalently, by the Law of Sines

Law of sines

In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

Trilinear coordinates

for the Nagel point are given by

or, equivalently, by the Law of Sines

Law of sines

In trigonometry, the law of sines is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles...

It is the isotomic conjugate of the Gergonne point.

Coordinates of the incenter

The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle 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

where

Trilinear coordinates
Trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...

for the incenter are given by

Barycentric coordinates
Barycentric coordinates (mathematics)
In geometry, the barycentric coordinate system is a coordinate system in which the location of a point is specified as the center of mass, or barycenter, of masses placed at the vertices of a simplex . Barycentric coordinates are a form of homogeneous coordinates...

for the incenter are given by

Equations for four circles

Let x : y : z be a variable point in trilinear coordinates

Trilinear coordinates

, and let u = cos²(A/2), v = cos²(B/2), w = cos²(C/2). The four circles described above are given by these equations:

Incircle:

A-excircle:

B-excircle:

C-excircle:

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

If the altitudes 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, i.e.

The distance between the circumcenter and the incenter is

where r is the incircle radius and R is the circumcircle radius. Thus the incircle radius is no larger than half the circumcircle radius (Euler's triangle inequality).

The product of the incircle radius and the circumcircle radius of a triangle with sides a, b, and c is

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.

External links

Derivation of formula for radius of incircle of a triangle

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
Cut-the-knot
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
Five Incircles Theorem at cut-the-knot
Cut-the-knot
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
Pairs of Incircles in a Quadrilateral at cut-the-knot
Cut-the-knot
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
An interactive Java applet for the incenter

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