Incircle and excircles

Summary

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]

Incircle and excircles of a triangle.
  Extended sides of triangle ABC
  Incircle (incenter at I)
  Excircles (excenters at JA, JB, JC)
  Internal angle bisectors
  External angle bisectors (forming the excentral triangle)

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]

Incircle and Incenter

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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

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

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

Trilinear coordinates

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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

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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

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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]  

Radius

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

where   is the semiperimeter.

The tangency points of the incircle divide the sides into segments of lengths   from  ,   from  , and   from  .[8]

See Heron's formula.

Distances to the vertices

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Denoting the incenter of   as  , the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation[9]  

Additionally,[10]  

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

Other properties

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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

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Distances between vertex and nearest touchpoints

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The distances from a vertex to the two nearest touchpoints are equal; for example:[11]  

Other properties

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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[13]  

Some relations among the sides, incircle radius, and circumcircle radius are:[14]  

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.[15]

Denoting the center of the incircle of   as  , we have[16]

 

and[17]: 121, #84   

The incircle radius is no greater than one-ninth the sum of the altitudes.[18]: 289 

The squared distance from the incenter   to the circumcenter   is given by[19]: 232   

and the distance from the incenter to the center   of the nine point circle is[19]: 232   

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).[19]: 233, Lemma 1 

Relation to area of the triangle

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The radius of the incircle is related to the area of the triangle.[20] 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.[21]

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:[citation needed]  

Gergonne triangle and point

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  Triangle ABC
  Incircle (incenter at I)
  Contact triangle TATBTC
  Lines between opposite vertices of ABC and TATBTC (concur at Gergonne point Ge)

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.[22]

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.[23]

The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.[24]

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,  

Excircles and excenters

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  Extended sides of ABC
  Incircle (incenter at I)
  Excircles (excenters at JA, JB, JC)
  Internal angle bisectors
  External angle bisectors (forming the excentral triangle)

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 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]

Trilinear coordinates of excenters

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While the incenter of   has trilinear coordinates  , the excenters have trilinears [citation needed]  

Exradii

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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.

Derivation of exradii formula

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Source:[25]

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

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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:[27]  

Other excircle properties

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The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.[28] The radius of this Apollonius circle is   where   is the incircle radius and   is the semiperimeter of the triangle.[29]

The following relations hold among the inradius  , the circumradius  , the semiperimeter  , and the excircle radii  ,  ,  :[14]  

The circle through the centers of the three excircles has radius  .[14]

If   is the orthocenter of  , then[14]  

Nagel triangle and Nagel point

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  Extended sides of triangle ABC
  Excircles of ABC (tangent at TA. TB, TC)
  Nagel/Extouch triangle TATBTC
  Splitters: lines connecting opposite vertices of ABC and TATBTC (concur at Nagel point N)

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 line segments  ,   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 X8).

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]

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Nine-point circle and Feuerbach point

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The nine-point circle is tangent to the incircle and excircles

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:[30][31]

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:[32]

... 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

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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]  

Equations for four circles

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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

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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]

Generalization to other polygons

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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.[37]

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]

See also

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Notes

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  1. ^ Kay (1969, p. 140)
  2. ^ a b Altshiller-Court (1925, p. 74)
  3. ^ a b c d e Altshiller-Court (1925, p. 73)
  4. ^ Kay (1969, p. 117)
  5. ^ a b Johnson 1929, p. 182.
  6. ^ a b Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine, accessed 2014-10-28.
  7. ^ Kay (1969, p. 201)
  8. ^ Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
  9. ^ Allaire, Patricia R.; Zhou, Junmin; Yao, Haishen (March 2012), "Proving a nineteenth century ellipse identity", Mathematical Gazette, 96: 161–165, doi:10.1017/S0025557200004277, S2CID 124176398.
  10. ^ Altshiller-Court, Nathan (1980), College Geometry, Dover Publications. #84, p. 121.
  11. ^ Mathematical Gazette, July 2003, 323-324.
  12. ^ Kay (1969, p. 203)
  13. ^ Johnson 1929, p. 189, #298(d).
  14. ^ a b c d "Bell, Amy, "Hansen's right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342" (PDF). Archived from the original (PDF) on 2021-08-31. Retrieved 2012-05-05.
  15. ^ Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
  16. ^ Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
  17. ^ Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
  18. ^ Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
  19. ^ a b c Franzsen, William N. (2011). "The distance from the incenter to the Euler line" (PDF). Forum Geometricorum. 11: 231–236. MR 2877263. Archived from the original (PDF) on 2020-12-05. Retrieved 2012-05-09..
  20. ^ Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
  21. ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
  22. ^ Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
  23. ^ 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
  24. ^ 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.
  25. ^ a b Altshiller-Court (1925, p. 79)
  26. ^ Kay (1969, p. 202)
  27. ^ 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.)
  28. ^ Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.
  29. ^ Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
  30. ^ Altshiller-Court (1925, pp. 103–110)
  31. ^ Kay (1969, pp. 18, 245)
  32. ^ Feuerbach, Karl Wilhelm; Buzengeiger, Carl Heribert Ignatz (1822), Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung (Monograph ed.), Nürnberg: Wiessner.
  33. ^ 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
  34. ^ Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
  35. ^ Johnson 1929, p. 187.
  36. ^ Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140.
  37. ^ Josefsson (2011, See in particular pp. 65–66.)

References

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  • 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
  • Johnson, Roger A. (1929), "X. Inscribed and Escribed Circles", Modern Geometry, Houghton Mifflin, pp. 182–194
  • Josefsson, Martin (2011), "More characterizations of tangential quadrilaterals" (PDF), Forum Geometricorum, 11: 65–82, MR 2877281, archived from the original (PDF) on 2016-03-04, retrieved 2023-03-14
  • 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.
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Interactive

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  • 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