Search results
Results from the WOW.Com Content Network
where r is the inradius and R is the circumradius of the triangle. Here the sign of the distances is taken to be negative if and only if the open line segment DX (X = F, G, H) lies completely outside the triangle. In the diagram, DF is negative and both DG and DH are positive. The theorem is named after Lazare Carnot (1753–1823).
Fuss' theorem gives a relation between the inradius r, the circumradius R and the distance x between the incenter I and the circumcenter O, for any bicentric quadrilateral. The relation is [1] [11] [22] + (+) =, or equivalently
Carnot's theorem (inradius, circumradius), describing a property of the incircle and the circumcircle of a triangle; Carnot's theorem (conics), describing a relation between triangles and conic sections; Carnot's theorem (perpendiculars), describing a property of certain perpendiculars on triangle sides; In physics:
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).
By Euler's theorem in geometry, the distance between the circumcenter O and the incenter I is ¯ = (), where r is the incircle radius and R is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler's triangle inequality), with equality only in the equilateral case.
Examples of cyclic quadrilaterals. In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle.This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic.
A complicated general formula is known for any number n of sides for the relation among the circumradius R, the inradius r, and the distance x between the circumcenter and the incenter. [5] Some of these for specific n are:
Tangential quadrilateral with inradius r. The four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites. If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, then that line passes through the ...