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]}^{: p. 182 } but not all polygons do; those that do are tangential polygons. See also tangent lines to circles.

See also: 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 .

The incenter is the point where the internal angle bisectors of meet.

The distance from vertex to the incenter is:^{[citation needed]}

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]}

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^{[citation needed]}

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.

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 , , and , and the sides opposite these vertices have corresponding lengths , , and , then the incenter is at^{[citation needed]}

The inradius of the incircle in a triangle with sides of length *, *, is given by^{[7]}

- where

See Heron's formula.

Denoting the incenter of as , the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation^{[8]}

Additionally,^{[9]}

where and are the triangle's circumradius and inradius respectively.

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]}

The distances from a vertex to the two nearest touchpoints are equal; for example:^{[10]}

Suppose the tangency points of the incircle divide the sides into lengths of and , and , and * and . Then the incircle has the radius*^{[11]}

and the area of the triangle is

If the altitudes from sides of lengths *, *, and are , , and *, then the inradius ** is one-third of the harmonic mean of these altitudes; that is,*^{[12]}

The product of the incircle radius * and the circumcircle radius of a triangle with sides **, *, and is^{[5]}^{: 189, #298(d) }

Some relations among the sides, incircle radius, and circumcircle radius are:^{[13]}

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.^{[14]}

Denoting the center of the incircle of as , we have^{[15]}

and^{[16]}^{: 121, #84 }

The incircle radius is no greater than one-ninth the sum of the altitudes.^{[17]}^{: 289 }

The squared distance from the incenter to the circumcenter is given by^{[18]}^{: 232 }

- ,

and the distance from the incenter to the center of the nine point circle is^{[18]}^{: 232 }

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).^{[18]}^{: 233, Lemma 1 }

"Inradius" redirects here. For the three-dimensional equivalent, see Inscribed sphere. |

The radius of the incircle is related to the area of the triangle.^{[19]} 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.^{[20]}

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 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:*^{[citation needed]}

- 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 * 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:*^{[citation needed]}

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.^{[21]}

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.^{[22]}

The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.^{[23]}

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,

An **excircle** or **escribed circle**^{[24]} 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 , 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 .^{[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]}^{: 182 }

While the incenter of * has trilinear coordinates , the excenters have trilinears , , and .*^{[citation needed]}

The radii of the excircles are called the **exradii**.

The exradius of the excircle opposite (so touching , centered at ) is^{[25]}^{[26]}

- where

See Heron's formula.

Click on *show* to view the contents of this section

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

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:^{[28]}

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.^{[29]} The radius of this Apollonius circle is where is the incircle radius and is the semiperimeter of the triangle.^{[30]}

The following relations hold among the inradius *, the circumradius , the semiperimeter **, and the excircle radii **, **, **:*^{[13]}

The circle through the centers of the three excircles has radius .^{[13]}

If * is the orthocenter of **, then*^{[13]}

Main article: Extouch triangle |

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 of the extouch ** is called the ***Mandart circle**.^{[citation needed]}

The three lines , and 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 (or triangle center *X*_{8}).

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]}

Main article: 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 concyclic points defined from the triangle. These nine points are:^{[31]}^{[32]}

- 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:^{[citation needed]}

- ... 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.

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^{[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 are given by^{[citation needed]}

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:*^{[33]}^{: 210–215 }

- Incircle:
*-*excircle:*-*excircle:*-*excircle:

Euler's theorem states that in a triangle:

where * and ** are the circumradius and inradius respectively, and ** is the distance between the circumcenter 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.*^{[34]}^{[35]}^{[36]}

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.^{[citation needed]}

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]}