Incircle and excircles explained

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

See also: Incenter.

Suppose

\triangle ABC has an incircle with radius r and center I .Let a be the length of \overline{BC} , b the length of \overline{AC} , and c the length of \overline{AB} .Also let T _{A} , T _{B} , and T _{C} be the touchpoints where the incircle touches \overline{BC} , \overline{AC} , and \overline{AB} .

Incenter

The incenter is the point where the internal angle bisectors of

\angle ABC, \angle BCA, and \angle BAC 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^[1] $$\backslash \; 1\; :\; 1\; :\; 1.$$

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$$\backslash \; a\; :\; b\; :\; c$$

where

a , b , and c are the lengths of the sides of the triangle, or equivalently (using the law of sines) by$$\backslash sin\; A\; :\; \backslash sin\; B\; :\; \backslash sin\; C$$

where

A , B , and C 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

(x _{a,y} _{a)} , (x _{b,y} _{b)} , and (x _{c,y} _{c)} , and the sides opposite these vertices have corresponding lengths a , b , and c , then the incenter is at$$\backslash left(\backslash frac,\; \backslash frac\backslash right)\; =\; \backslash frac.$$

Radius

The inradius

r of the incircle in a triangle with sides of length a , b , c is given by$$r\; =\; \backslash sqrt,$$

where

s = \tfrac 12 (a + b + c) is the semiperimeter.

The tangency points of the incircle divide the sides into segments of lengths

s - a from A , s - b from B , and s - c from C .^[2]

See Heron's formula.

Distances to the vertices

Denote the incenter of

\triangle ABC as I .

The distance from vertex

A to the incenter I is:$$\backslash overline\; =\; d(A,\; I)\; =\; c\; \backslash ,\; \backslash frac\; =\; b\; \backslash ,\; \backslash frac.$$

Derivation of the formula stated above

Use the Law of sines in the triangle

\triangle IAB .

We get

(\overline{AI)/() } = \frac.We have that \angle AIB = \pi - (A)/(2) - (B)/(= 2) (\pi)/(2) + (C)/(2) .

It follows that

\overline{AI} = c (\sin (B)/(2))/(\cos (C)/(2)) .

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^[3] $$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 1.$$

Additionally,^[4] $$\backslash overline\; \backslash cdot\; \backslash overline\; \backslash cdot\; \backslash overline\; =\; 4Rr^2,$$

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.^[1]

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:^[5] $$d\backslash left(A,\; T\_B\backslash right)\; =\; d\backslash left(A,\; T\_C\backslash right)\; =\; \backslash tfrac12(b\; +\; c\; -\; a)\; =\; s\; -\; a.$$

Other properties

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,$$r\; =\; \backslash frac.$$

The product of the incircle radius

r and the circumcircle radius R of a triangle with sides a , b , and c is$$rR\; =\; \backslash frac.$$

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.^[6]

The incircle radius is no greater than one-ninth the sum of the altitudes.^[7]

The squared distance from the incenter

I to the circumcenter O is given by^[8] $$\backslash overline^2\; =\; R(R\; -\; 2r)\; =\; \backslash frac\backslash left\; [\backslash frac\{a\backslash ,b\backslash ,c\backslash ,\}\{(a+b-c)\backslash ,(a-b+c)\backslash ,(-a+b+c)\}-1\; \backslash right\; ]$$

and the distance from the incenter to the center

N of the nine point circle is^[8]

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).^[8]

Relation to area of the triangle

The radius of the incircle is related to the area of the triangle.^[9] The ratio of the area of the incircle to the area of the triangle is less than or equal to

\pi / 3 \sqrt 3 , with equality holding only for equilateral triangles.^[10]

Suppose

\triangle ABC has an incircle with radius r and center I . Let a be the length of \overline{BC} , b the length of \overline{AC} , and c the length of \overline{AB} . Now, the incircle is tangent to \overline{AB} at some point T _{C} , and so\angle AT _{CI} is right. Thus, the radius T _{CI} is an altitude of \triangle IAB . Therefore,\triangle IAB has base length c and height r , and so has area \tfrac 12 cr .Similarly, \triangle IAC has area\tfrac 12 br and \triangle IBC has area\tfrac 12 ar .Since these three triangles decompose \triangle ABC , we see that the area \Delta of \triangle ABC is:$$\backslash Delta\; =\; \backslash tfrac12\; (a\; +\; b\; +\; c)r\; =\; sr,$$ and r = (\Delta)/(s) ,

where

\Delta is the area of \triangle ABC and s = \tfrac 12 (a + b + c) is its semiperimeter.

For an alternative formula, consider

\triangle IT _{CA} . This is a right-angled triangle with one side equal to r and the other side equal to r \cot \tfrac{A}{ 2 } . The same is true for \triangle IB'A . The large triangle is composed of six such triangles and the total area is: $$\backslash Delta\; =\; r^2\; \backslash left(\backslash cot\backslash tfrac\; +\; \backslash cot\backslash tfrac\; +\; \backslash cot\backslash tfrac\backslash right).$$

Gergonne triangle and point

The Gergonne triangle (of

\triangle ABC ) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite A is denoted T _{A} , etc.

This Gergonne triangle,

\triangle T _{AT} _{BT} _{C} , is also known as the contact triangle or intouch triangle of \triangle ABC . Its area is $$K\_T\; =\; K\backslash frac$$

where

K , r , and s are the area, radius of the incircle, and semiperimeter of the original triangle, and a , b , and c are the side lengths of the original triangle. This is the same area as that of the extouch triangle.^[11]

The three lines

AT _{A} , BT _{B} and CT _{C} intersect in a single point called the Gergonne point, denoted as G _{e} (or triangle center X₇). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.^[12]

The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.^[13]

Trilinear coordinates for the vertices of the intouch triangle are given by

Trilinear coordinates for the Gergonne point are given by $$\backslash sec^2\backslash tfrac\; :\; \backslash sec^2\backslash tfrac\; :\; \backslash sec^2\backslash tfrac,$$

or, equivalently, by the Law of Sines,$$\backslash frac\; :\; \backslash frac\; :\; \backslash frac.$$

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

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 . 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

\triangle ABC has trilinear coordinates 1 : 1 : 1 , the excenters have trilinears

Exradii

The radii of the excircles are called the exradii.

The exradius of the excircle opposite

A (so touching BC , centered at J _{A} ) is $$r\_a\; =\; \backslash frac\; =\; \backslash sqrt,$$ where s = \tfrac{ 1 }{ 2 }(a + b + c).

See Heron's formula.

Derivation of exradii formula

Source:

Let the excircle at side

AB touch at side AC extended at G , and let this excircle'sradius be r _{c} and its center be J _{c} . Then J _{c} G is an altitude of \triangle ACJ _{c} , so \triangle ACJ _{c} has area \tfrac 12 br _{c} . By a similar argument, \triangle BCJ _{c} has area \tfrac 12 ar _{c} and \triangle ABJ _{c} has area \tfrac 12 cr _{c} . Thus the area \Delta of triangle \triangle ABC is$$\backslash Delta\; =\; \backslash tfrac12\; (a\; +\; b\; -\; c)r\_c\; =\; (s\; -\; c)r\_c$$.

So, by symmetry, denoting

r as the radius of the incircle,$$\backslash Delta\; =\; sr\; =\; (s\; -\; a)r\_a\; =\; (s\; -\; b)r\_b\; =\; (s\; -\; c)r\_c$$.

By the Law of Cosines, we have$$\backslash cos\; A\; =\; \backslash frac$$

Combining this with the identity

\sin ^{2} A + \cos ^{2} A = 1 , we have$$\backslash sin\; A\; =\; \backslash frac$$

But

\Delta = \tfrac 12 bc \sin A , and so

which is Heron's formula.

Combining this with

sr = \Delta , we have$$r^2\; =\; \backslash frac\; =\; \backslash frac.$$

Similarly,

(s - a)r _{a} = \Delta 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:^[14] $$\backslash Delta\; =\; \backslash sqrt.$$

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.^[15] The radius of this Apollonius circle is

\tfrac{r ^{2} + s ^{2}{4r}} where r is the incircle radius and s is the semiperimeter of the triangle.^[16]

The following relations hold among the inradius

r , the circumradius R , the semiperimeter s , and the excircle radii r _{a} , r _{b} , r _{c} :^[17]

The circle through the centers of the three excircles has radius

2 R .^[17]

If

H is the orthocenter of \triangle ABC , then^[17]

Nagel triangle and Nagel point

See main article: Extouch triangle.

The Nagel triangle or extouch triangle of

\triangle 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 _{AT} _{BT} _{C} is also known as the extouch triangle of \triangle ABC . The circumcircle of the extouch \triangle T _{AT} _{BT} _{C} is called the Mandart circle(cf. Mandart inellipse).

The three line segments

\overline{AT _{A}} , \overline{BT _{B}} and \overline{CT _{C}} are called the splitters of the triangle; they each bisect the perimeter of the triangle,$$\backslash overline\; +\; \backslash overline\; =\; \backslash overline\; +\; \backslash overline\; =\; \backslash frac\backslash left(\backslash overline\; +\; \backslash overline\; +\; \backslash overline\; \backslash right).$$N _{a} (or triangle center X₈).

Trilinear coordinates for the vertices of the extouch triangle are given by

Trilinear coordinates for the Nagel point are given by $$\backslash csc^2\backslash tfrac\; :\; \backslash csc^2\backslash tfrac\; :\; \backslash csc^2\backslash tfrac,$$

or, equivalently, by the Law of Sines,$$\backslash frac\; :\; \backslash frac\; :\; \backslash frac.$$

The Nagel point is the isotomic conjugate of the Gergonne point.

Related constructions

Nine-point circle and Feuerbach point

See 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:

• 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:^[18]

... 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

\triangle ABC with the segments BC , CA , and AB are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle \triangle A'B'C' 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

\triangle A'B'C' are given by

Equations for four circles

Let

x:y:z be a variable point in trilinear coordinates, and let u = \cos ^{2\left} ( A/ 2 \right ) , v = \cos ^{2\left} ( B/ 2 \right ) , w = \cos ^{2\left} ( C/ 2 \right ) . The four circles described above are given equivalently by either of the two given equations:^[19]

• Incircle:$$\backslash begin$$

u^2 x^2 + v^2 y^2 + w^2 z^2 - 2vwyz - 2wuzx - 2uvxy &= 0 \\[4pt] &= 0\end

A -excircle:B -excircle:C -excircle:

Euler's theorem

Euler's theorem states that in a triangle:$$(R\; -\; r)^2\; =\; d^2\; +\; r^2,$$

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:$$\backslash left(R\; +\; r\_\backslash text\backslash right)^2\; =\; d\_\backslash text^2\; +\; r\_\backslash text^2,$$

where

r _{ex} is the radius of one of the excircles, and d _{ex} is the distance between the circumcenter and that excircle's center.^{[20] [21]}

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

References

• • • • • Clark . Kimberling . Triangle Centers and Central Triangles . Congressus Numerantium . 129 . 1998 . i-xxv,1–295 .
• Sándor . Kiss . The Orthic-of-Intouch and Intouch-of-Orthic Triangles . Forum Geometricorum . 6 . 2006 . 171–177 .

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

Notes and References

1. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of Triangle Centers
2. Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
3. .
4. . #84, p. 121.
5. Mathematical Gazette, July 2003, 323-324.
6. Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.
7. Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
8. Franzsen . William N. . Forum Geometricorum . 2877263 . 231–236 . The distance from the incenter to the Euler line . 11 . 2011 . 2012-05-09 . 2020-12-05 . https://web.archive.org/web/20201205220605/http://forumgeom.fau.edu/FG2011volume11/FG201126.pdf . dead . .
9. Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
10. Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
11. Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
12. 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
13. Dekov . Deko . Computer-generated Mathematics : The Gergonne Point . Journal of Computer-generated Euclidean Geometry . 2009 . 1 . 1 - 14 . dead . https://web.archive.org/web/20101105045604/http://www.dekovsoft.com/j/2009/01/JCGEG200901.pdf . 2010-11-05.
14. 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.)
15. http://forumgeom.fau.edu/FG2002volume2/FG200222.pdf Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182.
16. http://forumgeom.fau.edu/FG2003volume3/FG200320.pdf Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
17. Web site: Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342. . 2012-05-05 . 2021-08-31 . https://web.archive.org/web/20210831080348/https://forumgeom.fau.edu/FG2006volume6/FG200639.pdf . dead .
18. .
19. 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
20. Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
21. http://forumgeom.fau.edu/FG2001volume1/FG200120.pdf Emelyanov, Lev, and Emelyanova, Tatiana. "Euler’s formula and Poncelet’s porism", Forum Geometricorum 1, 2001: pp. 137–140.