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

\triangleABC

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

TA

,

TB

, and

TC

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

\angleABC,\angleBCA,and\angleBAC

meet.

The distance from vertex

A

to the incenter

I

is: d(A, I) = c \, \frac = b \, \frac.

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] \ 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\ 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\sin A : \sin B : \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

(xa,ya)

,

(xb,yb)

, and

(xc,yc)

, and the sides opposite these vertices have corresponding lengths

a

,

b

, and

c

, then the incenter is at \left(\frac, \frac\right) = \frac.

Radius

The inradius

r

of the incircle in a triangle with sides of length

a

,

b

,

c

is given byr = \sqrt,

where

s=\tfrac12(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

Denoting the incenter of

\triangleABC

as

I

, the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation[3] \frac + \frac + \frac = 1.

Additionally,[4] \overline \cdot \overline \cdot \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\left(A, T_B\right) = d\left(A, T_C\right) = \tfrac12(b + c - a) = s - a.

Other properties

If the altitudes from sides of lengths

a

,

b

, and

c

are

ha

,

hb

, and

hc

, then the inradius

r

is one-third of the harmonic mean of these altitudes; that is, r = \frac.

The product of the incircle radius

r

and the circumcircle radius

R

of a triangle with sides

a

,

b

, and

c

isrR = \frac.

Some relations among the sides, incircle radius, and circumcircle radius are:\begin ab + bc + ca &= s^2 + (4R + r)r, \\ a^2 + b^2 + c^2 &= 2s^2 - 2(4R + r)r.\end

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]

Denoting the center of the incircle of

\triangleABC

as

I

, we have[7]

\frac + \frac + \frac = 1

and[8] \overline \cdot \overline \cdot \overline = 4Rr^2.

The incircle radius is no greater than one-ninth the sum of the altitudes.[9]

The squared distance from the incenter

I

to the circumcenter

O

is given by[10] \overline^2 = R(R - 2r) = \frac\left [\frac{a\,b\,c\,}{(a+b-c)\,(a-b+c)\,(-a+b+c)}-1 \right ]

and the distance from the incenter to the center

N

of the nine point circle is[10] \overline = \tfrac12(R - 2r) < \tfrac12 R.

The incenter lies in the medial triangle (whose vertices are the midpoints of the sides).[10]

Relation to area of the triangle

The radius of the incircle is related to the area of the triangle.[11] The ratio of the area of the incircle to the area of the triangle is less than or equal to

\pi/3\sqrt3

, with equality holding only for equilateral triangles.[12]

Suppose

\triangleABC

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

TC

, and so

\angleATCI

is right. Thus, the radius

TCI

is an altitude of

\triangleIAB

. Therefore,

\triangleIAB

has base length

c

and height

r

, and so has area

\tfrac12cr

.Similarly,

\triangleIAC

has area

\tfrac12br

and

\triangleIBC

has area

\tfrac12ar

.Since these three triangles decompose

\triangleABC

, we see that the area

\Deltaof\triangleABC

is:\Delta = \tfrac12 (a + b + c)r = sr, and

r=

\Delta
s

,

where

\Delta

is the area of

\triangleABC

and

s=\tfrac12(a+b+c)

is its semiperimeter.

For an alternative formula, consider

\triangleITCA

. 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

\triangleIB'A

. The large triangle is composed of six such triangles and the total area is: \Delta = r^2 \left(\cot\tfrac + \cot\tfrac + \cot\tfrac\right).

Gergonne triangle and point

The Gergonne triangle (of

\triangleABC

) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite

A

is denoted

TA

, etc.

This Gergonne triangle,

\triangleTATBTC

, is also known as the contact triangle or intouch triangle of

\triangleABC

. Its area is K_T = K\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.[13]

The three lines

ATA

,

BTB

and

CTC

intersect in a single point called the Gergonne point, denoted as

Ge

(or triangle center X7). The Gergonne point lies in the open orthocentroidal disk punctured at its own center, and can be any point therein.[14]

The Gergonne point of a triangle has a number of properties, including that it is the symmedian point of the Gergonne triangle.[15]

Trilinear coordinates for the vertices of the intouch triangle are given by\begin T_A &=& 0 &:& \sec^2 \frac &:& \sec^2\frac \\[2pt] T_B &=& \sec^2 \frac &:& 0 &:& \sec^2\frac \\[2pt] T_C &=& \sec^2 \frac &:& \sec^2\frac &:& 0.\end

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

or, equivalently, by the Law of Sines,\frac : \frac : \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

\triangleABC

has trilinear coordinates

1:1:1

, the excenters have trilinears \begin J_A = & -1 &:& 1 &:& 1 \\ J_B = & 1 &:& -1 &:& 1 \\ J_C = & 1 &:& 1 &:& -1 \end

Exradii

The radii of the excircles are called the exradii.

The exradius of the excircle opposite

A

(so touching

BC

, centered at

JA

) is r_a = \frac = \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

rc

and its center be

Jc

. Then

JcG

is an altitude of

\triangleACJc

, so

\triangleACJc

has area

\tfrac12brc

. By a similar argument,

\triangleBCJc

has area

\tfrac12arc

and

\triangleABJc

has area

\tfrac12crc

. Thus the area

\Delta

of triangle

\triangleABC

is\Delta = \tfrac12 (a + b - c)r_c = (s - c)r_c.

So, by symmetry, denoting

r

as the radius of the incircle,\Delta = sr = (s - a)r_a = (s - b)r_b = (s - c)r_c.

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

Combining this with the identity

\sin2A+\cos2A=1

, we have\sin A = \frac

But

\Delta=\tfrac12bc\sinA

, and so\begin \Delta &= \tfrac14 \sqrt \\[5mu] &= \tfrac14 \sqrt \\[5mu] & = \sqrt,\end

which is Heron's formula.

Combining this with

sr=\Delta

, we haver^2 = \frac = \frac.

Similarly,

(s-a)ra=\Delta

gives\begin &r_a^2 = \frac \\[4pt] &\implies r_a = \sqrt.\end

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:[16] \Delta = \sqrt.

Other excircle properties

The circular hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle.[17] The radius of this Apollonius circle is

\tfrac{r2+s2}{4r}

where

r

is the incircle radius and

s

is the semiperimeter of the triangle.[18]

The following relations hold among the inradius

r

, the circumradius

R

, the semiperimeter

s

, and the excircle radii

ra

,

rb

,

rc

:[19] \begin r_a + r_b + r_c &= 4R + r, \\ r_a r_b + r_b r_c + r_c r_a &= s^2, \\ r_a^2 + r_b^2 + r_c^2 &= \left(4R + r\right)^2 - 2s^2.\end

The circle through the centers of the three excircles has radius

2R

.[19]

If

H

is the orthocenter of

\triangleABC

, then[19] \begin r_a + r_b + r_c + r &= \overline + \overline + \overline + 2R, \\ r_a^2 + r_b^2 + r_c^2 + r^2 &= \overline^2 + \overline^2 + \overline^2 + (2R)^2.\end

Nagel triangle and Nagel point

See main article: Extouch triangle.

The Nagel triangle or extouch triangle of

\triangleABC

is denoted by the vertices

TA

,

TB

, and

TC

that are the three points where the excircles touch the reference

\triangleABC

and where

TA

is opposite of

A

, etc. This

\triangleTATBTC

is also known as the extouch triangle of

\triangleABC

. The circumcircle of the extouch

\triangleTATBTC

is called the Mandart circle.

The three line segments

\overline{ATA}

,

\overline{BTB}

and

\overline{CTC}

are called the splitters of the triangle; they each bisect the perimeter of the triangle,\overline + \overline = \overline + \overline = \frac\left(\overline + \overline + \overline \right).

Na

(or triangle center X8).

Trilinear coordinates for the vertices of the extouch triangle are given by\begin T_A &=& 0 &:& \csc^2\frac &:& \csc^2\frac \\[2pt] T_B &=& \csc^2\frac &:& 0 &:& \csc^2\frac \\[2pt] T_C &=& \csc^2\frac &:& \csc^2\frac &:& 0\end

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

or, equivalently, by the Law of Sines,\frac : \frac : \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:

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:[20]

... 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

\triangleABC

with the segments

BC

,

CA

, and

AB

are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle

\triangleA'B'C'

are given by\begin A' &=& 0 &:& 1 &:& 1 \\[2pt] B' &=& 1 &:& 0 &:& 1 \\[2pt] C' &=& 1 &:& 1 &:& 0\end

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

\triangleA'B'C'

are given by\begin A' &=& -1 &:& 1 &:& 1\\[2pt] B' &=& 1 &:& -1 &:& 1 \\[2pt] C' &=& 1 &:& 1 &:& -1\end

Equations for four circles

Let

x:y:z

be a variable point in trilinear coordinates, and let

u=\cos2\left(A/2\right)

,

v=\cos2\left(B/2\right)

,

w=\cos2\left(C/2\right)

. The four circles described above are given equivalently by either of the two given equations:[21]

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

A

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

B

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

C

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

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:\left(R + r_\text\right)^2 = d_\text^2 + r_\text^2,

where

rex

is the radius of one of the excircles, and

dex

is the distance between the circumcenter and that excircle's center.[22] [23]

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

References

External links

Interactive

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. Allaire, Patricia R.; Zhou, Junmin; and Yao, Haishen, "Proving a nineteenth century ellipse identity", Mathematical Gazette 96, March 2012, 161-165.
  8. Altshiller-Court, Nathan. College Geometry, Dover Publications, 1980.
  9. Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
  10. Franzsen . William N. . Forum Geometricorum . 2877263 . 231–236 . The distance from the incenter to the Euler line . 11 . 2011. .
  11. Coxeter, H.S.M. "Introduction to Geometry 2nd ed. Wiley, 1961.
  12. Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theorem 4.1.
  13. Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ContactTriangle.html
  14. 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
  15. 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.
  16. 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.)
  17. 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.
  18. http://forumgeom.fau.edu/FG2003volume3/FG200320.pdf Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.
  19. http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", Forum Geometricorum 6, 2006, 335–342.
  20. .
  21. 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
  22. Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
  23. 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.