“Circumradius”的概念、定义、习题解题方法及翻译-数学辞典

单词

Circumradius

释义

Circumradius

The radius of a Triangle's Circumcircle or of a Polyhedron's Circumsphere, denoted . For a Triangle,

(1)

where the side lengths of the Triangle are

, and

This equation can also be expressed in terms of the Radii of the three mutually tangentCircles centered at the Triangle's Vertices. Relabeling the diagram forthe Soddy Circles with Vertices , , and and the radii , , and, and using

			(2)
			(3)
			(4)

then gives

(5)

is the Circumcenter and

is the triangle Centroid, then

(6)

(7)

(8)

(9)

(10)

(Johnson 1929, pp. 189-191). Let

be the distance between Inradius

and circumradius

. Then

(11)

(12)

(Mackay 1886-87). These and many other identities are given in Johnson (1929, pp. 186-190).

For an Archimedean Solid, expressing the circumradius in terms of the Inradius and Midradius gives

			(13)
			(14)

for an Archimedean Solid.

See also Carnot's Theorem, Circumcircle, Circumsphere

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.

Mackay, J. S. ``Historical Notes on a Geometrical Theorem and its Developments [18th Century].'' Proc. Edinburgh Math. Soc. 5, 62-78, 1886-1887.

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