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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).
Further, combining these formulas yields: [25] ... where and are the circumradius and inradius respectively, and is the distance between the ...
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]
Another formula for the distance x between the centers of the incircle and the circumcircle is due to the American mathematician Leonard Carlitz (1907–1999). It states that [24] = where r and R are the inradius and the circumradius respectively, and
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:
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
Because the square of the area of an integer triangle is rational, the square of its circumradius is also rational, as is the square of the inradius. The ratio of the inradius to the circumradius of an integer triangle is rational, equaling 4 T 2 / s a b c {\displaystyle 4T^{2}/sabc} for semiperimeter s and area T .