Incircle and excircles of a triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle 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}
All regular polygons have incircles tangent to all sides, 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 r and center I. Let a be the length of BC, b the length of AC, and c the length of AB. Also let be the touchpoints where the incircle touches BC, AC, and AB.
Incenter
The incenter is the point where the internal angle bisectors of meet.
The distance from vertex A to the incenter I 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[6]
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 r of the incircle in a triangle with sides of length a, b, c is given by
- , where .[7]
See Heron's formula.
Distances to the vertices
Denoting the incenter of triangle ABC as I, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation[8]
Additionally,[9]
where R and r 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.[6]
Incircle and its radius properties
Distances between vertex and nearest touchpoints
The distances from a vertex to the two nearest touchpoints are equal:
Other 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[10]
and the area of the triangle is
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; that is,
The product of the incircle radius r and the circumcircle radius R of a triangle with sides a, b, and c is[5]^{:p. 189, #298(d)}
Some relations among the sides, incircle radius, and circumcircle radius are:[12]
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.[13]
Denoting the center of the incircle of triangle ABC as I, we have[14]
and[15]^{:p.121,#84}
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.[16] Thus for example for vertex B and adjacent tangencies T_{A} and T_{C},
The incircle radius is no greater than one-ninth the sum of the altitudes.[17]^{:p. 289}
The squared distance from the incenter I to the circumcenter O is given by[18]^{:p.232}
and the distance from the incenter to the center N of the nine point circle is[18]^{:p.232}
The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).[18]^{:p.233, Lemma 1}
Relation to area of the triangle
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 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, T_{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 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 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:
Gergonne triangle and point
The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. The touchpoint opposite A is denoted T_{A}, etc.
This Gergonne triangle T_{A}T_{B}T_{C} is also known as the contact triangle or intouch triangle of ABC. Its area is
where , , are the area, radius of the incircle and semiperimeter of the original triangle, and , , are the side lengths of the original triangle. This is the same area as that of the extouch triangle.[21]
The three lines AT_{A}, BT_{B} and CT_{C} intersect in a single point called the 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.[22]
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.[23]
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[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 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}
Trilinear coordinates of excenters
While triangle ABC's incenter has trilinear coordinates its excenters (the centers of its excircles) have trilinears , and
Exradii
The radii of the excircles are called the exradii.
The exradius of the excircle opposite A (so touching BC, centered at ) is
See Heron's formula.
Derivation of exradii formula
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 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
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:[28]
Other excircle properties
The circular hull of the excircles is internally tangent to each of the excircles, and thus is an Apollonius circle.[29] The radius of this Apollonius circle is where r is the incircle radius and s is the semiperimeter of the triangle.[30]
The following relations hold among the inradius r, the circumradius R, the semiperimeter s, and the excircle radii r_{a}, r_{b}, r_{c}:[12]
The circle through the centers of the three excircles has radius 2R.[12]
If H is the orthocenter of triangle ABC, then[12]
Nagel triangle and Nagel point
The Nagel triangle or extouch triangle of ABC is denoted by the vertices T_{A}, T_{B} and T_{C} that are the three points where the excircles touch the reference triangle ABC and where T_{A} is opposite of A, etc. This triangle T_{A}T_{B}T_{C} is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle T_{A}T_{B}T_{C} is called the Mandart circle.
The three lines AT_{A}, BT_{B} and CT_{C} 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 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.
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).[31][32]
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... (Feuerbach 1822)
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 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
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 x : y : z be a variable point in trilinear coordinates, and let u = cos^{2}(A/2), v = cos^{2}(B/2), w = cos^{2}(C/2). The four circles described above are given equivalently by either of the two given equations:[33]^{:p. 210–215}
- Incircle:
- A-excircle:
- B-excircle:
- C-excircle:
Euler's theorem
Euler's theorem states that in a triangle:
where R and r are the circumradius and inradius respectively, and d is the distance between the circumcenter and the incenter.
For excircles the equation is similar:
where r_{ex} is the radius of one of the excircles, and d_{ex} is the distance between the circumcenter and this excircle's center.[34][35][36]
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—one that is tangent to each side—is called a tangential polygon.
See also
Notes
- Kay (1969, p. 140)
- Altshiller-Court (1925, p. 74)
- Altshiller-Court (1925, p. 73)
- Kay (1969, p. 117)
- Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929).
- Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine, accessed 2014-10-28.
- Kay (1969, p. 201)
- Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette, 96: 161–165.
- Altshiller-Court, Nathan (1980), College Geometry, Dover Publications. #84, p. 121.
- Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
- Kay (1969, p. 203)
- Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
- Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
- Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
- Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
- Mathematical Gazette, July 2003, 323-324.
- Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
- Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. MR 2877263..
- Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
- Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
- Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
- Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", Forum Geometricorum 6 (2006), 57–70. http://forumgeom.fau.edu/FG2006volume6/FG200607index.html
- Dekov, Deko (2009). "Computer-generated Mathematics : The Gergonne Point" (PDF). Journal of Computer-generated Euclidean Geometry. 1: 1–14. Archived from the original (PDF) on 2010-11-05.
- Altshiller-Court (1925, p. 74)
- Altshiller-Court (1925, p. 79)
- Kay (1969, p. 202)
- Altshiller-Court (1925, p. 79)
- Baker, Marcus, "A collection of formulae for the area of a plane triangle," Annals of Mathematics, part 1 in vol. 1(6), January 1885, 134-138. (See also part 2 in vol. 2(1), September 1885, 11-18.)
- Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.
- Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
- Altshiller-Court (1925, pp. 103–110)
- Kay (1969, pp. 18,245)
- Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books
- Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
- Johnson, R. A. Modern Geometry, Houghton Mifflin, Boston, 1929: p. 187.
- Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140.
References
- Altshiller-Court, Nathan (1925), College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.), New York: Barnes & Noble, LCCN 52013504
- Kay, David C. (1969), College Geometry, New York: Holt, Rinehart and Winston, LCCN 69012075
- Kimberling, Clark (1998). "Triangle Centers and Central Triangles". Congressus Numerantium (129): i–xxv, 1–295.
- Kiss, Sándor (2006). "The Orthic-of-Intouch and Intouch-of-Orthic Triangles". Forum Geometricorum (6): 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