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Difference between revisions of "Incircle"

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[[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with [[incenter]] ''I'', with [[angle bisector]]s (red), [[incircle]] (blue), and [[inradius|inradii]] (green)]]
 
  
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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. Quadrilaterals that do have an incircle are called tangential quadrilaterals.  
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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]], where the [[incircle]] is the largest circle that can be inscribed in the polygon. The [[Incenter]] can be constructed by drawing the intersection of angle bisectors.
  
 
==Formulas==
 
==Formulas==
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*The radius of an incircle of a triangle (the inradius) with sides <math>a,b,c</math> and area <math>K</math>  is <math>\frac{2K}{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>
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*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>.
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*The [[area]] of any [[triangle]] is <math>r \times s,</math> where <math>s</math> is the [[Semiperimeter]] of the [[triangle]].
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*For any polygon with an incircle, <math>K=sr</math>, where <math>K</math> is the area, <math>s</math> is the semiperimeter, and <math>r</math> is the inradius.
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*The formula above can be simplified with Heron's Formula, yielding <math>r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.</math>
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*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>.
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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>.
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*The formula for the [[semiperimeter]] is <math>s=\frac{a+b+c}{2}</math>.
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*The [[area]] of the [[triangle]] by [[Heron's Formula]] is <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math>.
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==See also==
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*[[Circumradius]]
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*[[Inradius]]
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*[[Kimberling center]]
  
 
[[Category:Geometry]]
 
[[Category:Geometry]]
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Click here to learn about the orthocenter, and Line's Tangent

Latest revision as of 17:38, 9 March 2023


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, where the incircle is the largest circle that can be inscribed in the polygon. The Incenter can be constructed by drawing the intersection of angle bisectors.

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 area of any triangle is $r \times s,$ where $s$ is the Semiperimeter of the triangle.
  • The formula above can be simplified with Heron's Formula, yielding $r = \sqrt{\frac{(s-a)(s-b)(s-c)}{s}}.$
  • 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 $c$.

See also

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

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