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]].  
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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]]. 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>
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*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 semiperimeter, and <math>r</math> is the inradius.
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*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.
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*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>.
  
*And area of the triangle by Heron is <math>A^2=s(s-a)(s-b)(s-c)</math>.
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*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]]
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Click here to learn about the orthrocenter, and Line's Tangent

Latest revision as of 00:49, 18 July 2020

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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 $a,b,c$ and area $A$ is $r =$ $\frac{2A}{a+b+c}$
  • The radius of an incircle of a right triangle (the inradius) with legs $a,b$ and hypotenuse $c$ is $r=\frac{ab}{a+b+c}=\frac{a+b-c}{2}$.
  • For any polygon with an incircle, $A=sr$, where $A$ is the area, $s$ is the semi perimeter, and $r$ is the inradius.
  • The coordinates of the incenter (center of incircle) are $(\dfrac{aA_x+bB_x+cC_x}{a+b+c}, \dfrac{aA_y+bB_y+cC_y}{a+b+c})$, if the coordinates of each vertex are $A(A_x, A_y)$, $B(B_x, B_y)$, and $C(C_x, C_y)$, the side opposite of $A$ has length $a$, the side opposite of $B$ has length $b$, and the side opposite of $C$ has length $c$.
  • The formula for the semiperimeter is $s=\frac{a+b+c}{2}$.

See also

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

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