Incircle and excircles

 ( Circles Defined For A Triangle ) 

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

$$r\; =\; \backslash frac\{1\}\{\backslash dfrac\{1\}\{h\_a\}\; +\; \backslash dfrac\{1\}\{h\_b\}\; +\; \backslash dfrac\{1\}\{h\_c\}\}.$$

The product of the incircle radius $r$ and the circumcircle radius $R$ of a triangle with sides $a$, $b$, and $c$ is

$$rR\; =\; \backslash frac\{abc\}\{2(a\; +\; b\; +\; c)\}.$$

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

$$\backslash begin\{align\}$$

    ab + bc + ca &=  s^2 +  (4R + r)r, \\
 a^2 + b^2 + c^2 &= 2s^2 - 2(4R + r)r.
     

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.Kodokostas, Dimitrios, "Triangle Equalizers", Mathematics Magazine 83, April 2010, pp. 141-146.

The incircle radius is no greater than one-ninth the sum of the altitudes.Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.

The squared distance from the incenter $I$ to the circumcenter $O$ is given by.

$$\backslash overline\{OI\}^2\; =\; R(R\; -\; 2r)\; =\; \backslash frac\{a\backslash ,b\backslash ,c\backslash ,\}\{a+b+c\}\backslash left\; \backslash frac\{a\backslash ,b\backslash ,c\backslash ,\}\{(a+b-c)\backslash ,(a-b+c)\backslash ,(-a+b+c)\}-1$$

and the distance from the incenter to the center $N$ of the nine point circle is

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

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

Now, the incircle is tangent to $\backslash overline\{AB\}$ at some point $T\_C$, and so $\backslash angle\; AT\_CI$ is right. Thus, the radius $T\_CI$ is an altitude of $\backslash triangle\; IAB$.

Therefore, $\backslash triangle\; IAB$ has base length $c$ and height $r$, and so has area $\backslash tfrac12\; cr$.

Similarly, $\backslash triangle\; IAC$ has area $\backslash tfrac12\; br$ and $\backslash triangle\; IBC$ has area $\backslash tfrac12\; ar$.

Since these three triangles decompose $\backslash triangle\; ABC$, we see that the area $\backslash Delta\; \backslash text\{\; of\}\; \backslash triangle\; ABC$ is:

$$\backslash Delta\; =\; \backslash tfrac12\; (a\; +\; b\; +\; c)r\; =\; sr,$$

and $r\; =\; \backslash frac\{\backslash Delta\}\{s\},$
     

where $\backslash Delta$ is the area of $\backslash triangle\; ABC$ and $s\; =\; \backslash tfrac12(a\; +\; b\; +\; c)$ is its semiperimeter.

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

$$\backslash Delta\; =\; r^2\; \backslash left(\backslash cot\backslash tfrac\{A\}\{2\}\; +\; \backslash cot\backslash tfrac\{B\}\{2\}\; +\; \backslash cot\backslash tfrac\{C\}\{2\}\backslash right).$$

]]

The Gergonne triangle (of $\backslash triangle\; ABC$) 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, $\backslash triangle\; T\_AT\_BT\_C$, is also known as the contact triangle or intouch triangle of $\backslash triangle\; ABC$. Its area is

$$K\_T\; =\; K\backslash frac\{2r^2\; s\}\{abc\}$$

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. Weisstein, Eric W. "Contact Triangle." /ref>

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.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

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

$$\backslash begin\{array\}\{ccccccc\}$$

 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.
     

Trilinear coordinates for the Gergonne point are given by

$$\backslash sec^2\backslash tfrac\{A\}\{2\}\; :\; \backslash sec^2\backslash tfrac\{B\}\{2\}\; :\; \backslash sec^2\backslash tfrac\{C\}\{2\},$$

or, equivalently, by the Law of Sines,

$$\backslash frac\{bc\}\{b\; +\; c\; -\; a\}\; :\; \backslash frac\{ca\}\{c\; +\; a\; -\; b\}\; :\; \backslash frac\{ab\}\{a\; +\; b\; -\; c\}.$$

]]

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 extended side. 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.

$$\backslash begin\{array\}\{rrcrcr\}$$

 J_A = & -1 &:& 1 &:& 1 \\
 J_B = & 1 &:& -1 &:& 1 \\
 J_C = & 1 &:& 1 &:& -1
     

The exradius of the excircle opposite $A$ (so touching $BC$, centered at $J\_A$) is

$$r\_a\; =\; \backslash frac\{rs\}\{s\; -\; a\}\; =\; \backslash sqrt\{\backslash frac\{s(s\; -\; b)(s\; -\; c)\}\{s\; -\; a\}\},$$ where $s\; =\; \backslash tfrac\{1\}\{2\}(a\; +\; b\; +\; c).$

See Heron's formula.

Let the excircle at side $AB$ touch at side $AC$ extended at $G$, and let this excircle's radius be $r\_c$ and its center be $J\_c$. Then $J\_c\; G$ is an altitude of $\backslash triangle\; ACJ\_c$, so $\backslash triangle\; ACJ\_c$ has area $\backslash tfrac12\; br\_c$. By a similar argument, $\backslash triangle\; BCJ\_c$ has area $\backslash tfrac12\; ar\_c$ and $\backslash triangle\; ABJ\_c$ has area $\backslash tfrac12\; cr\_c$. Thus the area $\backslash Delta$ of triangle $\backslash triangle\; ABC$ 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

$$\backslash cos\; A\; =\; \backslash frac\{b^2\; +\; c^2\; -\; a^2\}\{2bc\}$$

Combining this with the identity $\backslash sin^2\; \backslash !\; A\; +\; \backslash cos^2\; \backslash !\; A\; =\; 1$, we have

$$\backslash sin\; A\; =\; \backslash frac\{\backslash sqrt\{-a^4\; -\; b^4\; -\; c^4\; +\; 2a^2\; b^2\; +\; 2b^2\; c^2\; +\; 2\; a^2\; c^2\}\}\{2bc\}$$

But $\backslash Delta\; =\; \backslash tfrac12\; bc\; \backslash sin\; A$, and so

$$\backslash begin\{align\}$$

 \Delta &= \tfrac14 \sqrt{-a^4 - b^4 - c^4 + 2a^2b^2 + 2b^2 c^2 + 2 a^2 c^2} \\[5mu]
        &= \tfrac14 \sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)} \\[5mu]
        & = \sqrt{s(s - a)(s - b)(s - c)},
     

\end{align}

which is Heron's formula.

Combining this with $sr\; =\; \backslash Delta$, we have

$$r^2\; =\; \backslash frac\{\backslash Delta^2\}\{s^2\}\; =\; \backslash frac\{(s\; -\; a)(s\; -\; b)(s\; -\; c)\}\{s\}.$$

Similarly, $(s\; -\; a)r\_a\; =\; \backslash Delta$ gives

$$\backslash begin\{align\}$$

 &r_a^2 = \frac{s(s - b)(s - c)}{s - a} \\[4pt]
 &\implies r_a = \sqrt{\frac{s(s - b)(s - c)}{s - a}}.
     

\end{align}

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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: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.)

$$\backslash Delta\; =\; \backslash sqrt\{r\; r\_a\; r\_b\; r\_c\}.$$

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Other excircle properties

The circular convex hull of the excircles is internally tangent to each of the excircles and is thus an Apollonius circle. Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", Forum Geometricorum 2, 2002: pp. 175-182. The radius of this Apollonius circle is $\backslash tfrac\{r^2\; +\; s^2\}\{4r\}$ where $r$ is the incircle radius and $s$ is the semiperimeter of the triangle. Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", Forum Geometricorum 3, 2003, 187-195.

The following relations hold among the inradius $r$, the circumradius $R$, the semiperimeter $s$, and the excircle radii $r\_a$, $r\_b$, $r\_c$:

$$\backslash begin\{align\}$$

             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}

The circle through the centers of the three excircles has radius $2R$.

If $H$ is the orthocenter of $\backslash triangle\; ABC$, then

$$\backslash begin\{align\}$$

         r_a + r_b + r_c + r &= \overline{AH} + \overline{BH} + \overline{CH} + 2R, \\
 r_a^2 + r_b^2 + r_c^2 + r^2 &= \overline{AH}^2 + \overline{BH}^2 + \overline{CH}^2 + (2R)^2.
     

\end{align}

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Nagel triangle and Nagel point

[[File:Extouch Triangle and Nagel Point.svg|right|thumb|300px|

]]

The Nagel triangle or extouch triangle of $\backslash triangle\; ABC$ is denoted by the vertices $T\_A$, $T\_B$, and $T\_C$ that are the three points where the excircles touch the reference $\backslash triangle\; ABC$ and where $T\_A$ is opposite of $A$, etc. This $\backslash triangle\; T\_AT\_BT\_C$ is also known as the extouch triangle of $\backslash triangle\; ABC$. The circumcircle of the extouch $\backslash triangle\; T\_AT\_BT\_C$ is called the Mandart circle (cf. Mandart inellipse).

The three line segments $\backslash overline\{AT\_A\}$, $\backslash overline\{BT\_B\}$ and $\backslash overline\{CT\_C\}$ are called the splitters of the triangle; they each bisect the perimeter of the triangle,

$$\backslash overline\{AB\}\; +\; \backslash overline\{BT\_A\}\; =\; \backslash overline\{AC\}\; +\; \backslash overline\{CT\_A\}\; =\; \backslash frac\{1\}\{2\}\backslash left(\; \backslash overline\{AB\}\; +\; \backslash overline\{BC\}\; +\; \backslash overline\{AC\}\; \backslash right).$$

The splitters intersect in a single point, the triangle's Nagel point $N\_a$ (or triangle center X₈).

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

$$\backslash begin\{array\}\{ccccccc\}$$

 T_A &=& 0 &:& \csc^2\frac{B}{2} &:& \csc^2\frac{C}{2} \\[2pt]
 T_B &=& \csc^2\frac{A}{2} &:& 0 &:& \csc^2\frac{C}{2} \\[2pt]
 T_C &=& \csc^2\frac{A}{2} &:& \csc^2\frac{B}{2} &:& 0
     

\end{array}

Trilinear coordinates for the Nagel point are given by

$$\backslash csc^2\backslash tfrac\{A\}\{2\}\; :\; \backslash csc^2\backslash tfrac\{B\}\{2\}\; :\; \backslash csc^2\backslash tfrac\{C\}\{2\},$$

or, equivalently, by the Law of Sines,

$$\backslash frac\{b\; +\; c\; -\; a\}\{a\}\; :\; \backslash frac\{c\; +\; a\; -\; b\}\{b\}\; :\; \backslash frac\{a\; +\; b\; -\; c\}\{c\}.$$

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

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Related constructions

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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 perpendicular 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 circles 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.

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Incentral and excentral triangles

The points of intersection of the interior angle bisectors of $\backslash triangle\; ABC$ with the segments $BC$, $CA$, and $AB$ are the vertices of the incentral triangle. Trilinear coordinates for the vertices of the incentral triangle $\backslash triangle\; A\text{'}B\text{'}C\text{'}$ are given by

$$\backslash begin\{array\}\{ccccccc\}$$

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

\end{array}

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 $\backslash triangle\; A\text{'}B\text{'}C\text{'}$ are given by

$$\backslash begin\{array\}\{ccrcrcr\}$$

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

\end{array}

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Equations for four circles

Let $x:y:z$ be a variable point in trilinear coordinates, and let $u=\backslash cos^2\backslash left\; (\; A/2\; \backslash right\; )$, $v=\backslash cos^2\backslash left\; (\; B/2\; \backslash right\; )$, $w=\backslash cos^2\backslash left\; (\; C/2\; \backslash right\; )$. The four circles described above are given equivalently by either of the two given equations:Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions /ref>

• Incircle:$$\backslash begin\{align\}$$

 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}

• $A$-excircle:$$\backslash begin\{align\}$$

 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}

• $B$-excircle:$$\backslash begin\{align\}$$

 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}

• $C$-excircle:$$\backslash begin\{align\}$$

 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}

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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:

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

where $r\_\backslash text\{ex\}$ is the radius of one of the excircles, and $d\_\backslash text\{ex\}$ is the distance between the circumcenter and that excircle's center.Nelson, Roger, "Euler's triangle inequality via proof without words", Mathematics Magazine 81(1), February 2008, 58-61. Emelyanov, Lev, and Emelyanova, Tatiana. "Euler's formula and Poncelet's porism", Forum Geometricorum 1, 2001: pp. 137–140.

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Generalization to other polygons

Some (but not all) 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.

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See also

• Triangle conic

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Notes

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External links

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

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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

Categories: Circles Defined For A Triangle