Incircle and excircles

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. but not all polygons do; those that do are tangential polygons. 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 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
Barycentric coordinates
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
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 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 is the semiperimeter.
The tangency points of the incircle divide the sides into segments of lengths and
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 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
The distances from a vertex to the two nearest touchpoints are equal; for example:
Other properties
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,
The product of the incircle radius and the circumcircle radius of a triangle with sides and 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.
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 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 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.
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.
The three lines , and intersect in a single point called the Gergonne point, denoted as (or triangle center X7). 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 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
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
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. 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 :
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 of the extouch is called the Mandart circle.
The three line segments , 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 (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 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:
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:
... 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:
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 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.
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.
See also
Triangle conic
Notes
References
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
Five Incircles Theorem at cut-the-knot
Pairs of Incircles in a Quadrilateral at cut-the-knot
An interactive Java applet for the incenter
Circles defined for a triangle