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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).
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).
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. [7] [8] The distance between O and the orthocenter H is [9] [10]
The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is ( s − a ) ( s − b ) {\displaystyle (s-a)(s-b)} where a, b are the legs. For quadrilaterals
1.3 Inradius theorems. ... from combining Heron's formula for the area of a triangle in terms of the sides with the area formula ... and the triangle's circumradius ...
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:
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 .
A triangle with sides <, semiperimeter = (+ +), area, altitude opposite the longest side, circumradius, inradius, exradii,, tangent to ,, respectively, and medians,, is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.