Incircle and excircles

[[File:Incircle and Excircles.svg|right|thumb|300px|Incircle and excircles of a triangle.

]]

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 .

:<math display=block>\ 1 : 1 : 1.</math>

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

:<math display=block>a : b : c</math>

where <math>a</math>, <math>b</math>, and <math>c</math> are the lengths of the sides of the triangle, or equivalently (using the law of sines) by

:<math display=block>\sin A : \sin B : \sin C</math>

where <math>A</math>, <math>B</math>, and <math>C</math> 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 <math>(x_a,y_a)</math>, <math>(x_b,y_b)</math>, and <math>(x_c,y_c)</math>, and the sides opposite these vertices have corresponding lengths <math>a</math>, <math>b</math>, and <math>c</math>, then the incenter is at

:<math display=block>

\left(\frac{a x_a + b x_b + c x_c}{a + b + c}, \frac{a y_a + b y_b + c y_c}{a + b + c}\right)

= \frac{a\left(x_a, y_a\right) + b\left(x_b, y_b\right) + c\left(x_c, y_c\right)}{a + b + c}.

</math>

Radius

The inradius <math>r</math> of the incircle in a triangle with sides of length <math>a</math>, <math>b</math>, <math>c</math> is given by

:<math display=block>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s,</math>

where <math>s = \tfrac12(a + b + c)</math> is the semiperimeter (see Heron's formula).

The tangency points of the incircle divide the sides into segments of lengths <math>s-a</math> from <math>A</math>, <math>s-b</math> from <math>B</math>, and <math>s-c</math> from <math>C</math> (see Tangent lines to a circle).

Distances to the vertices

Denote the incenter of <math>\triangle ABC</math> as <math>I</math>.

The distance from vertex <math>A</math> to the incenter <math>I</math> is:

:<math display=block>

\overline{AI} = d(A, I)

= c \, \frac{\sin\frac{B}{2{\cos\frac{C}{2

= b \, \frac{\sin\frac{C}{2{\cos\frac{B}{2.

</math>

Derivation of the formula stated above

Use the Law of sines in the triangle <math>\triangle IAB</math>.

We get <math>\frac{\overline{AI{\sin \frac{B}{2 = \frac{c}{\sin \angle AIB}</math>.

We have that <math>\angle AIB = \pi - \frac{A}{2} - \frac{B}{2} = \frac{\pi}{2} + \frac{C}{2}</math>.

It follows that <math>\overline{AI} = c \ \frac{\sin \frac{B}{2{\cos \frac{C}{2</math>.

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

:<math display=block>\frac{\overline{IA} \cdot \overline{IA{\overline{CA} \cdot \overline{AB + \frac{\overline{IB} \cdot \overline{IB{\overline{AB} \cdot \overline{BC + \frac{\overline{IC} \cdot \overline{IC{\overline{BC} \cdot \overline{CA = 1.</math>

Additionally,

:<math display=block>\overline{IA} \cdot \overline{IB} \cdot \overline{IC} = 4Rr^2,</math>

where <math>R</math> and <math>r</math> 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.

:<math display=block>d\left(A, T_B\right) = d\left(A, T_C\right) = \tfrac12(b + c - a) = s - a.</math>

Other properties

If the altitudes from sides of lengths <math>a</math>, <math>b</math>, and <math>c</math> are <math>h_a</math>, <math>h_b</math>, and <math>h_c</math>, then the inradius <math>r</math> is one third the harmonic mean of these altitudes; that is,

:<math display=block> r = \frac{1}{\dfrac{1}{h_a} + \dfrac{1}{h_b} + \dfrac{1}{h_c.</math>

The product of the incircle radius <math>r</math> and the circumcircle radius <math>R</math> of a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> is

:<math display=block>rR = \frac{abc}{2(a + b + c)}.</math>

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

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

The squared distance from the incenter <math>I</math> to the circumcenter <math>O</math> is given by

:<math display=block>\overline{OI}^2 = R(R - 2r) = \frac{a\,b\,c\,}{a+b+c}\left [\frac{a\,b\,c\,}{(a+b-c)\,(a-b+c)\,(-a+b+c)}-1 \right ]</math>

and the distance from the incenter to the center <math>N</math> of the nine point circle is The ratio of the area of the incircle to the area of the triangle is less than or equal to <math>\pi \big/ 3\sqrt3</math>,

with equality holding only for equilateral triangles.

Suppose <math>\triangle ABC</math> has an incircle with radius <math>r</math> and center <math>I</math>. Let <math>a</math> be the length of <math>\overline{BC}</math>, <math>b</math> the length of <math>\overline{AC}</math>, and <math>c</math> the length of <math>\overline{AB}</math>.

Now, the incircle is tangent to <math>\overline{AB}</math> at some point <math>T_C</math>, and so <math>\angle AT_CI</math> is right. Thus, the radius <math>T_CI</math> is an altitude of <math>\triangle IAB</math>.

Therefore, <math>\triangle IAB</math> has base length <math>c</math> and height <math>r</math>, and so has area <math>\tfrac12 cr</math>.

thumb|[[Proof without words that the area of a triangle equals the product of its inradius and its semiperimeter]]

Similarly, <math>\triangle IAC</math> has area <math>\tfrac12 br</math> and <math>\triangle IBC</math> has area <math>\tfrac12 ar</math>.

Since these three triangles decompose <math>\triangle ABC</math>, we see that the area <math>\Delta \text{ of} \triangle ABC</math> is:

:<math display=block>\Delta = \tfrac12 (a + b + c)r = sr,</math>

and <math>r = \frac{\Delta}{s},</math>

where <math>\Delta</math> is the area of <math>\triangle ABC</math> and <math>s = \tfrac12(a + b + c)</math> is its semiperimeter.

For an alternative formula, consider <math>\triangle IT_CA</math>. This is a right-angled triangle with one side equal to <math>r</math> and the other side equal to <math>r \cot \tfrac{A}{2}</math>. The same is true for <math>\triangle IB'A</math>. The large triangle is composed of six such triangles and the total area is:

:<math display=block>\Delta = r^2 \left(\cot\tfrac{A}{2} + \cot\tfrac{B}{2} + \cot\tfrac{C}{2}\right).</math>

Gergonne triangle and point

[[File:Intouch Triangle and Gergonne Point.svg|right|frame|

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The Gergonne triangle (of <math>\triangle ABC</math>) is defined by connecting the three touchpoints of the incircle on the three sides. The touchpoint opposite <math>A</math> is denoted <math>T_A</math>, etc.

This Gergonne triangle, <math>\triangle T_AT_BT_C</math>, is also known as the contact triangle or intouch triangle of <math>\triangle ABC</math>. Its area is

:<math display=block>K_T = K\frac{2r^2 s}{abc}</math>

where <math>K</math>, <math>r</math>, and <math>s</math> are the area, radius of the incircle, and semiperimeter of the original triangle, and <math>a</math>, <math>b</math>, and <math>c</math> are the side lengths of the original triangle. This is the same area as that of the extouch triangle.

The three lines <math>AT_A</math>, <math>BT_B</math>, and <math>CT_C</math> intersect in a single point called the Gergonne point, denoted as <math>G_e</math> (or triangle center X<sub>7</sub>). 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

:<math display=block>\begin{array}{ccccccc}

T_A &=& 0 &:& \sec^2 \frac{B}{2} &:& \sec^2\frac{C}{2} \\[2pt]

T_B &=& \sec^2 \frac{A}{2} &:& 0 &:& \sec^2\frac{C}{2} \\[2pt]

T_C &=& \sec^2 \frac{A}{2} &:& \sec^2\frac{B}{2} &:& 0.

\end{array}</math>

Trilinear coordinates for the Gergonne point are given by

:<math display=block>\sec^2\tfrac{A}{2} : \sec^2\tfrac{B}{2} : \sec^2\tfrac{C}{2},</math>

or, equivalently, by the law of cosines,

:<math display=block>\frac{bc}{b + c - a} : \frac{ca}{c + a - b} : \frac{ab}{a + b - c}.</math>

Excircles and excenters

[[File:Incircle and Excircles.svg|right|thumb|300px|

]]

An excircle or escribed circle

:<math display=block>r_a = \frac{rs}{s - a} = \sqrt{\frac{s(s - b)(s - c)}{s - a,</math> where <math>s = \tfrac{1}{2}(a + b + c).</math>

See Heron's formula.

Derivation of exradii formula

Source:

:<math display=block>\Delta = \sqrt{r r_a r_b r_c}.</math>

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 <math>\tfrac{r^2 + s^2}{4r}</math> where <math>r</math> is the incircle radius and <math>s</math> is the semiperimeter of the triangle.

The following relations hold among the inradius <math>r</math>, the circumradius <math>R</math>, the semiperimeter <math>s</math>, and the excircle radii <math>r_a</math>, <math>r_b</math>, <math>r_c</math>:

:<math display=block>\begin{align}

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{align}</math>

The circle through the centers of the three excircles has radius <math>2R</math>.

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.

The incircle may be described as the pedal circle of the incenter. The locus of points whose pedal circles are tangent to the nine-point circle is known as the McCay cubic.

Incentral and excentral triangles

The points of intersection of the interior angle bisectors of <math>\triangle ABC</math> with the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle <math>\triangle A'B'C'</math> are given by

:<math display=block>\begin{array}{ccccccc}

A' &=& 0 &:& 1 &:& 1 \\[2pt]

B' &=& 1 &:& 0 &:& 1 \\[2pt]

C' &=& 1 &:& 1 &:& 0

\end{array}</math>

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 <math>\triangle A'B'C'</math> are given by

:<math display=block>\begin{array}{ccrcrcr}

A' &=& -1 &:& 1 &:& 1\\[2pt]

B' &=& 1 &:& -1 &:& 1 \\[2pt]

C' &=& 1 &:& 1 &:& -1

\end{array}</math>

Equations for four circles

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

Incircle:<math display=block>\begin{align}

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

{\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2 &= 0

\end{align}</math>

<math>A</math>-excircle:<math display=block>\begin{align}

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

{\textstyle \pm\sqrt{-x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2 &= 0

\end{align}</math>

<math>B</math>-excircle:<math display=block>\begin{align}

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

{\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{-y\vphantom{t\cos\tfrac{B}{2} \pm \sqrt{z}\cos\tfrac{C}{2 &= 0

\end{align}</math>

<math>C</math>-excircle:<math display=block>\begin{align}

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

{\textstyle \pm\sqrt{x}\cos\tfrac{A}{2} \pm \sqrt{y\vphantom{t\cos\tfrac{B}{2} \pm \sqrt{-z}\cos\tfrac{C}{2 &= 0

\end{align}</math>

Euler's theorem

Euler's theorem states that in a triangle:

:<math display=block>(R - r)^2 = d^2 + r^2,</math>

where <math>R</math> and <math>r</math> are the circumradius and inradius respectively, and <math>d</math> is the distance between the circumcenter and the incenter.

For excircles the equation is similar:

:<math display=block>\left(R + r_\text{ex}\right)^2 = d_\text{ex}^2 + r_\text{ex}^2,</math>

where <math>r_\text{ex}</math> is the radius of one of the excircles, and <math>d_\text{ex}</math> 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.

Generalization to topological triangles

If topological triangles are considered, it is still possible to define an inscribed circle. It is no longer described as tangent to all sides, since the topological triangle might not be differentiable everywhere. Rather, it is defined as a circle whose center has the same minimal distance to each side. It has been proven that all topological triangles have an inscribed circle.

Notes

References

External links

Derivation of formula for radius of incircle of a triangle, MATHalino

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

Barycentric coordinates

Cartesian coordinates

Radius

Distances to the vertices

Derivation of the formula stated above

Other properties

Other properties

Gergonne triangle and point

Excircles and excenters

Derivation of exradii formula

Other excircle properties

Incentral and excentral triangles

Equations for four circles

Euler's theorem

Generalization to other polygons

Generalization to topological triangles

See also

Notes

References

External links

Interactive