Inscribed circle



In geometry, the incircle or inscribed circle of a polygon is the largest circle contained in the polygon; it touches (is tangent to) the many sides. The center of the incircle is called the polygon's incenter.

An excircle or escribed circle of the polygon is a circle lying outside the polygon, tangent to one of its sides and tangent to the extensions of the other two. Every polygon has many distinct excircles, each tangent to one of the polygons sides.

The center of the incircle can be found as the intersection of the many 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. From this, it follows that the center of the incircle together with the many 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 A 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


 * $$A = \frac{1}{4}\sqrt{(a+b+c)(a-b+c)(b-c+a)(c-a+b)}= \sqrt{s(s-a)(s-b)(s-c)}$$

where $$s = \frac{(a + b + c)}{2}$$ is the semiperimeter.

Radius
The radius of the incircle (AKA the inradius) is


 * $$r= \frac{2A}{P} = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.= \frac{\sqrt{\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2}}}{a+b+c}= s\frac{sin(\frac{360}{n})}{2(1+cos(\frac{180}{n})+sin(\frac{180}{n}))}= s\frac{n}{(6n-12)tan(\frac{180}{n})}$$

Diameter
The diameter of the incircle is:
 * $$d= \frac{4A}{P}= \frac{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}{a+b+c}= s\frac{sin(\frac{360}{n})}{1+cos(\frac{180}{n})+sin(\frac{180}{n})}= s\frac{n}{(3n-6)tan(\frac{180}{n})}$$

Others
The excircle at side a has radius


 * $$\frac{2A}{c-a+b}.$$

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


 * $$\frac{2A}{a-b+c}$$

and


 * $$\frac{2A}{b-c+a}.$$

From these formulas we see in particular that the excircles are always larger than the incircle, and that the largest excircle is the one attached to the longest side.

Area

 * $$a= \frac{4A^2}{P^2}\pi$$

The area of the incircle is:
 * $$a= \frac{\frac{(a^2+b^2+c^2)^2}{4}-\frac{a^4+b^4+c^4}{2}}{(a+b+c)^2}\pi$$


 * $$a= s^2\pi\frac{n^2}{(6n-12)^2tan^2(\frac{180}{n})}$$

From a right triangle:
 * $$a= s^2\pi\frac{sin^2(\frac{360}{n})}{4(1+cos(\frac{180}{n})+sin(\frac{180}{n}))^2}$$

Perimeter

 * $$p= \frac{4A}{P}\pi$$

The perimeter of the incircle is:
 * $$p= \frac{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}{a+b+c}\pi$$


 * $$p= s\pi\frac{n}{(3n-6)tan(\frac{180}{n})}$$

From a right triangle:
 * $$p= s\pi\frac{sin(\frac{360}{n})}{1+cos(\frac{180}{n})+sin(\frac{180}{n})}$$

Incircles of polygons
The inradius of a regular n-sided polygon is:
 * $$r= \frac{s}{2tan(\frac{180}{n})}$$

The diameter of incircle:
 * $$d= s\frac{2}{2tan(\frac{180}{n})}$$

Area and Perimeter
Incircle area:
 * $$a= s^2\frac{\pi}{4tan^2(\frac{180}{n})}$$

Incircle perimeter:
 * $$p= s\frac{2\pi}{2tan(\frac{180}{n})}$$



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 point of a triangle is the symmedian point of its contact triangle. Denoting the three vertices of the triangle by A, B and C and the three points where the incircle touches the triangle by TA, TB and TC (where TA is opposite of A, etc.), the triangle TATBTC is known as the contact triangle or Gergonne triangle of ABC. The incircle of ABC is the circumcircle of TATBTC. The three lines ATA, BTB and CTC intersect in a single point, the triangle's Gergonne point G.

The contact triangle is also called the intouch triangle, and the touchpoints of the excircle with segments BC,CA,AC are the vertices of the extouch triangle. The Gergonne triangle is also called the excentral triangle, and the points of intersection of the interior angle bisectors of ABC with the segments BC,CA,AB are the vertices of the incentral triangle.

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

or, equivalently, by the Law of Sines, $$\frac{bc}{b+ c - a} : \frac{ca}{c + a-b} : \frac{ab}{a+b-c}$$.

Coordinates of the incenter
The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices. (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 opposite sides of the triangle have lengths $$a$$, $$b$$, and $$c$$, then the incenter is at
 * $$\bigg(\frac{a x_a+b x_b+c x_c}{P},\frac{a y_a+b y_b+c y_c}{P}\bigg) = \frac{a}{P}(x_a,y_a)+\frac{b}{P}(x_b,y_b)+\frac{c}{P}(x_c,y_c)$$.


 * Trilinear coordinates for the incenter are 1 : 1 : 1.
 * Barycentric coordinates for the incenter are 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: u2x2 + v2y2 + w2z2 - 2(vwyz - wuzx - uvxy) = 0
 * A-excircle: u2x2 + v2y2 + w2z2 - 2(vwyz + wuzx + uvxy) = 0
 * B-excircle: u2x2 + v2y2 + w2z2 + 2(vwyz - wuzx + uvxy) = 0
 * C-excircle: u2x2 + v2y2 + w2z2 + 2(vwyz + wuzx - uvxy) = 0