# Difference between revisions of "Incircle"

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− | An '''incircle''' of a [[convex]] [[polygon]] is a [[circle]] which is inside the figure and [[tangent line | tangent]] to each side. Every [[triangle]] and [[regular polygon]] has a unique incircle, but in general polygons with 4 or more sides (such as non-[[square (geometry) | square]] [[rectangle]]s) do not have an incircle. A quadrilateral that does have an incircle is called a [[Tangential Quadrilateral]]. | + | An '''incircle''' of a [[convex]] [[polygon]] is a [[circle]] which is inside the figure and [[tangent line | tangent]] to each side. Every [[triangle]] and [[regular polygon]] has a unique incircle, but in general polygons with 4 or more sides (such as non-[[square (geometry) | square]] [[rectangle]]s) do not have an incircle. A quadrilateral that does have an incircle is called a [[Tangential Quadrilateral]]. For a triangle, the center of the incircle is the [[Incenter]]. |

==Formulas== | ==Formulas== | ||

− | *The radius of an incircle of a triangle (the inradius) with sides <math>a,b,c</math> and area <math>A</math> is <math>\frac{2A}{a+b+c}</math> | + | *The radius of an incircle of a triangle (the inradius) with sides <math>a,b,c</math> and area <math>A</math> is <math>r =</math> <math>\frac{2A}{a+b+c}</math> |

*The radius of an incircle of a right triangle (the inradius) with legs <math>a,b</math> and hypotenuse <math>c</math> is <math>r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}</math>. | *The radius of an incircle of a right triangle (the inradius) with legs <math>a,b</math> and hypotenuse <math>c</math> is <math>r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}</math>. | ||

− | *For any polygon with an incircle, <math>A=sr</math>, where <math>A</math> is the area, <math>s</math> is the | + | *For any polygon with an incircle, <math>A=sr</math>, where <math>A</math> is the area, <math>s</math> is the semi perimeter, and <math>r</math> is the inradius. |

+ | *The coordinates of the incenter (center of incircle) are <math>(\dfrac{aA_x+bB_x+cC_x}{a+b+c}, \dfrac{aA_y+bB_y+cC_y}{a+b+c})</math>, if the coordinates of each vertex are <math>A(A_x, A_y)</math>, <math>B(B_x, B_y)</math>, and <math>C(C_x, C_y)</math>, the side opposite of <math>A</math> has length <math>a</math>, the side opposite of <math>B</math> has length <math>b</math>, and the side opposite of <math>C</math> has length <math>c</math>. | ||

*The formula for the semiperimeter is <math>s=\frac{a+b+c}{2}</math>. | *The formula for the semiperimeter is <math>s=\frac{a+b+c}{2}</math>. | ||

− | * | + | *The area of the triangle by [[Heron's Formula]] is <math>A^2=s(s-a)(s-b)(s-c)</math>. |

==See also== | ==See also== | ||

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[[Category:Geometry]] | [[Category:Geometry]] | ||

+ | Click here to learn about the orthrocenter, and Line's Tangent |

## Latest revision as of 00:49, 18 July 2020

*This article is a stub. Help us out by expanding it.*

An **incircle** of a convex polygon is a circle which is inside the figure and tangent to each side. Every triangle and regular polygon has a unique incircle, but in general polygons with 4 or more sides (such as non- square rectangles) do not have an incircle. A quadrilateral that does have an incircle is called a Tangential Quadrilateral. For a triangle, the center of the incircle is the Incenter.

## Formulas

- The radius of an incircle of a triangle (the inradius) with sides and area is
- The radius of an incircle of a right triangle (the inradius) with legs and hypotenuse is .
- For any polygon with an incircle, , where is the area, is the semi perimeter, and is the inradius.
- The coordinates of the incenter (center of incircle) are , if the coordinates of each vertex are , , and , the side opposite of has length , the side opposite of has length , and the side opposite of has length .

- The formula for the semiperimeter is .

- The area of the triangle by Heron's Formula is .

## See also

Click here to learn about the orthrocenter, and Line's Tangent