Incircle and excircles of a triangle

A triangle (black) with incircle (blue), incentre (I), excircles (orange), excentres (J_A,J_B,J_C), internal angle bisectors (red) and external angle bisectors (green)

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is 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 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 and the external bisectors of the other two. 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.

See also Tangent lines to circles.

Relation to area of the triangle

The radii of the in- and excircles are closely related to the area of the triangle. Let K be the triangle's area and let a, b and c, be the lengths of its sides. By Heron's formula, the area of the triangle is

$\begin{align} K & {} = \frac{1}{4}\sqrt{(P)(a-b+c)(b-c+a)(c-a+b)} \\ & {} = \sqrt{s(s-a)(s-b)(s-c)} \end{align}$

where $s= \tfrac{1}{2}(a+b+c)$ is the semiperimeter and P = 2s is the perimeter.

The radius of the incircle (also known as the inradius, r ) is

$r = \frac{2K}{P} = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.$

Thus, the area K of a triangle may be found by multiplying the inradius by the semiperimeter:

$\displaystyle K=rs.$

The radii in the excircles are called the exradii. The excircle at side a has radius

$r_a = \frac{2K}{c-a+b} = \sqrt{\frac{s (s-b)(s-c)}{s-a}}.$

Similarly the radii of the excircles at sides b and c are respectively

$r_b = \frac{2K}{a-b+c} = \sqrt{\frac{s (s-a)(s-c)}{s-b}}$

and

$r_c = \frac{2K}{b-c+a} = \sqrt{\frac{s (s-a)(s-b)}{s-c}}.$

From these formulas 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 with Heron's area formula yields the result that^[1]:#4

$K=\sqrt{rr_ar_br_c}.$

A triangle, ΔABC, with incircle (blue), incenter (blue, I), contact triangle (red, ΔT_aT_bT_c) and Gergonne point (green, Ge)

Nine-point circle and Feuerbach point

The circle tangent to all three of the excircles as well as the incircle is known as the nine-point circle. The point where the nine-point circle touches the incircle is known as the Feuerbach point.

Gergonne triangle and point

The Gergonne triangle of ABC is denoted by the vertices T_A, T_B and T_C that are the three points where the incircle touches the reference triangle ABC and where T_A is opposite of A, etc. This triangle T_AT_BT_C is also known as the contact triangle or intouch triangle of ABC. The incircle of ABC is the circumcircle of T_AT_BT_C. The three lines AT_A, BT_B and CT_C intersect in a single point, the triangle's Gergonne point Ge - X(7). Interestingly, the Gergonne point of a triangle is the symmedian point of its Gergonne triangle. For a full set of properties of the Gergonne point see.^[2]

The touchpoints of the excircle with segments BC,CA,AB are the vertices of the extouch triangle. The points of intersection of the interior angle bisectors of ABC with the segments BC,CA,AB are the vertices of the incentral triangle.

Nagel triangle and point

The Nagel triangle of ABC is denoted by the vertices X_A, X_B and X_C that are the three points where the excircles touches the reference triangle ABC and where X_A is opposite of A, etc. This triangle X_AX_BX_C is also known as the extouch triangle of ABC. The circumcircle of the extouch triangle X_AX_BX_C is called the Mandart circle. The three lines AX_A, BX_B and CX_C intersect in a single point, the triangle's Nagel point Na - X(8).

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

• $A-\text{vertex}= 0 : \sec^2 \left(\frac{B}{2}\right) :\sec^2\left(\frac{C}{2}\right)$
• $B-\text{vertex}= \sec^2 \left(\frac{A}{2}\right):0:\sec^2\left(\frac{C}{2}\right)$
• $C-\text{vertex}= \sec^2 \left(\frac{A}{2}\right) :\sec^2\left(\frac{B}{2}\right):0$

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

• $A-\text{vertex} = 0 : \csc^2\left(\frac{B}{2}\right) : \csc^2\left(\frac{C}{2}\right)$
• $B-\text{vertex} = \csc^2\left(\frac{A}{2}\right) : 0 : \csc^2\left(\frac{C}{2}\right)$
• $C-\text{vertex} = \csc^2\left(\frac{A}{2}\right) : \csc^2\left(\frac{B}{2}\right) : 0$

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

• $\ A-\text{vertex} = 0 : 1 : 1$
• $\ B-\text{vertex} = 1 : 0 : 1$
• $\ C-\text{vertex} = 1 : 1 : 0$

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

• $\ A-\text{vertex}= -1 : 1 : 1$
• $\ B-\text{vertex}= 1 : -1 : 1$
• $\ C-\text{vertex}= 1 : -1 : -1$

Trilinear coordinates for the Gergonne point are given by

$\sec^2\left(\frac{A}{2}\right) : \sec^2 \left(\frac{B}{2}\right) : \sec^2\left(\frac{C}{2}\right)$,

or, equivalently, by the Law of Sines,

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

Trilinear coordinates for the Nagel point are given by

$\csc^2\left(\frac{A}{2}\right) : \csc^2 \left(\frac{B}{2}\right) : \csc^2\left(\frac{C}{2}\right)$,

or, equivalently, by the Law of Sines,

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

It is the isotomic conjugate of the Gergonne point.

Coordinates of the incenter

The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle 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

$\bigg(\frac{a x_a+b x_b+c x_c}{P},\frac{a y_a+b y_b+c y_c}{P}\bigg) = \frac{a(x_a,y_a)+b(x_b,y_b)+c(x_c,y_c)}{P}$

where $\ P = a + b + c.$

• Trilinear coordinates for the incenter are given by

$\ 1 : 1 : 1.$

• Barycentric coordinates for the incenter are given by

$\ a : b : c.$

Equations for four circles

Let x : y : z be a variable point in trilinear coordinates, and let u = cos²(A/2), v = cos²(B/2), w = cos²(C/2). The four circles described above are given by these equations:

• Incircle:

$\ u^2x^2+v^2y^2+w^2z^2-2vwyz-2wuzx-2uvxy=0$

• A-excircle:

$\ u^2x^2+v^2y^2+w^2z^2-2vwyz+2wuzx+2uvxy=0$

• B-excircle:

$\ u^2x^2+v^2y^2+w^2z^2+2vwyz-2wuzx+2uvxy=0$

• C-excircle:

$\ u^2x^2+v^2y^2+w^2z^2+2vwyz+2wuzx-2uvxy=0$

Other incircle properties

Suppose the tangency points of the incircle divide the sides into lengths of x and y, y and z, and z and x. Then the incircle has the radius^[3]

$r = \sqrt{\frac{xyz}{x+y+z}}.$

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, i.e.

$r = \frac{1}{h_a^{-1}+h_b^{-1}+h_c^{-1}}.$

The distance between the circumcenter and the incenter is $\sqrt{R(R-2r)},$ where r is the incircle radius and R is the circumcircle radius.^[4] Thus the incircle radius is no larger than half the circumcircle radius (Euler's triangle inequality).^[4]

The product of the incircle radius and the circumcircle radius of a triangle with sides a, b, and c is^[5] $\frac{abc}{2(a+b+c)}.$

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]

See also

• Altitude (triangle)
• Circumscribed circle
• Ex-tangential quadrilateral
• Inscribed sphere
• Power of a point
• Steiner inellipse
• Tangential quadrilateral
• Triangle center

References

1. ^ 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.)
2. ^ Dekov, Deko (2009). "Computer-generated Mathematics : The Gergonne Point". Journal of Computer-generated Euclidean Geometry 1: 1–14.. http://www.dekovsoft.com/j/2009/01/JCGEG200901.pdf. 
3. ^ Chu, Thomas, The Pentagon, Spring 2005, p. 45, problem 584.
4. ^ ^{a b} Nelson, Roger, "Euler's triangle inequality via proof without words," Mathematics Magazine 81(1), February 2008, 58-61.
5. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover, 2007 (orig. 1929), p. 189, #298(d).
6. ^ Kodokostas, Dimitrios, "Triangle Equalizers," Mathematics Magazine 83, April 2010, pp. 141-146.

• Clark Kimberling, "Triangle Centers and Central Triangles," Congressus Numerantium 129 (1998) i-xxv and 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
• Weisstein, Eric W., "Incircle" from MathWorld.

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