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In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
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 formula can be derived from the law of sines. 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.
All triangles can have an incircle, but not all quadrilaterals do. ... where s is the semiperimeter and r is the inradius. Another formula is [7] = ...
In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter D to the sides of an arbitrary triangle ABC is + + = +, where r is the inradius and R is the circumradius of the triangle.
Using the usual notations for a triangle (see the figure at the upper right), where a, b, c are the lengths of the three sides, A, B, C are the vertices opposite those three respective sides, α, β, γ are the corresponding angles at those vertices, s is the semiperimeter, that is, s = a + b + c / 2 , and r is the radius of the inscribed circle, the law of cotangents states that
A formula for the area of bicentric ... The inradius can also be expressed in terms of ... which is the analog of Euler's theorem for triangles for bicentric ...
A triangle is a polygon with three corners and ... Its radius is called the inradius. ... is a formula for finding the area of a triangle from the lengths of its ...