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The inradius of the incircle in a triangle with sides of length , , is given by [7] = () (), where s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} is the semiperimeter. The tangency points of the incircle divide the sides into segments of lengths s − a {\displaystyle s-a} from A {\displaystyle A} , s − b {\displaystyle s-b ...
This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals , circumscribing quadrilaterals , and circumscriptible ...
All radii, once calculated, are divided by 6.957 × 10 8 to convert from m to R ☉.. AD radius determined from angular diameter and distance =, (/) =, = D is multiplied by 3.0857 × 10 19 to convert from kpc to m
The inradius is = () (). The law of cotangents gives the cotangents of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius. The length of the internal bisector of the angle opposite the side of length a is [1]
If r and R are the inradius and the circumradius respectively, then the area K satisfies the inequalities [14] 4 r 2 ≤ K ≤ 2 R 2 . {\displaystyle \displaystyle 4r^{2}\leq K\leq 2R^{2}.} There is equality on either side only if the quadrilateral is a square .
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the excircles; they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form an orthocentric system. [26]
Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles – a right kite.A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential.