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Wikipedia / Incircle and excircles

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.

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 (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.

Incircle and Incenter

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Incenter

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

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

Barycentric coordinates

The barycentric coordinates are the lengths of the sides of the triangle, or equivalently (using the law of sines are the angles at the three vertices.

Cartesian coordinates

The Cartesian coordinates, then the incenter is at

Distances to the vertices

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Derivation of the formula stated above

Use the Law of sines.

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The equality with the second expression is obtained the same way.

The distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation

Additionally, are the triangle's circumradius and inradius respectively.

Other properties

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.

Incircle and its radius properties

Distances between vertex and nearest touchpoints

: is the semiperimeter.

(see Tangent lines to a circle).

Radius

is given by is the semiperimeter (see Heron's formula).

Relation to area of the triangle

The radius of the incircle is related to the area of the triangle., with equality holding only for equilateral triangles.

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is an altitude.

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is its semiperimeter.

. The large triangle is composed of six such triangles and the total area is:

Other properties

If the altitudes is one third the harmonic mean of these altitudes; that is,

and the circumcircle 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 (the center of its incircle). There are either one, two, or three of these for any given triangle.

The incircle radius is no greater than one-ninth the sum of the altitudes.

to the circumcenter is given by of the nine point circle is

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).

Gergonne triangle and point

The Gergonne triangle, etc.

, is also known as the contact triangle or intouch triangle are the side lengths of the original triangle. This is the same area as that of the extouch triangle.

intersect in a single point called the Gergonne point (or triangle center ). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.

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 cosines

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.

, for example) and the external bisectors of the other two. The center of this excircle is called the excenter, or the excenter. 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 has trilinear coordinates, the excenters have trilinears

Exradii

The radii of the excircles are called the exradii.

) is

See Heron's formula.

Derivation of exradii formula

Source:

By the Law of Cosines

which is Heron's formula.

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 of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle. is the semiperimeter of the triangle.

:

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is the orthocenter, then

Nagel triangle and Nagel point

The Nagel triangle or extouch triangle is also known as the extouch triangle. The circumcircle is called the Mandart circle (cf. Mandart inellipse).

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 (or triangle center ).

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 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:

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 triangle center at which the incircle and the nine-point circle touch is called the Feuerbach point.

The incircle may be described as the pedal circle of the incenter. The locus of points whose pedal circles are tangent to the nine-point circle is known as the McCay cubic.

Incentral and excentral triangles

are 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 are given by

Equations for four circles

be a variable point in trilinear coordinates. The four circles described above are given equivalently by either of the two given equations:

Euler's theorem

Euler's theorem is the distance between the circumcenter and the incenter.

is the distance between the circumcenter and that excircle's center.

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.

Generalization to topological triangles

If topological triangles are considered, it is still possible to define an inscribed circle. It is no longer described as tangent to all sides, since the topological triangle might not be differentiable everywhere. Rather, it is defined as a circle whose center has the same minimal distance to each side. It has been proven that all topological triangles have an inscribed circle.

See also

Notes

References

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