Difference between revisions of "Incircle"
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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>. | *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. | *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>( | + | *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>. | ||
Revision as of 00:11, 30 May 2019
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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
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 .
and area 
is 
and hypotenuse 
is 
.
, where 
is the semiperimeter, and 
is the inradius.
, if the coordinates of each vertex are 
, 
, and 
, the side opposite of 
, 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
.
.
