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. | + | 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> | + | *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 right triangle (the inradius) with legs <math>a,b</math> and hypotenuse <math>c</math> is <math>\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> | + | *For any polygon with an incircle, <math>A=sr</math>, where <math>A</math> is the area, <math>s</math> is the semiperimeter, 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>. | ||
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+ | *The formula for the semiperimeter is <math>s=\frac{a+b+c}{2}</math>. | ||
+ | |||
+ | *And area of the triangle by Heron is <math>A^2=s(s-a)(s-b)(s-c)</math>. | ||
+ | |||
+ | ==See also== | ||
+ | *[[Circumradius]] | ||
+ | *[[Inradius]] | ||
+ | *[[Kimberling center]] | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
+ | Click here to learn about the orthocenter, and Line's Tangent |
Revision as of 13:01, 23 November 2019
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 semiperimeter, 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 .
- And area of the triangle by Heron is .
See also
Click here to learn about the orthocenter, and Line's Tangent