| 释义 |
InradiusThe radius of a Triangle's Incircle or of a Polyhedron's Insphere, denoted  . For a Triangle,
where  is the Area of the Triangle,  ,  , and  are the side lengths,  is theSemiperimeter, and  is the Circumradius (Johnson 1929, p. 189).
Equation (1) can be derived easily using Trilinear Coordinates. Since the Incenter is equally spaced from allthree sides, its trilinear coordinates are 1:1:1, and its exact trilinear coordinates are  . The ratio  of the exacttrilinears to the homogeneous coordinates is given by
 | (3) |
 in this case,
 | (4) |
Other equations involving the inradius include
 | (5) |
 | (6) |
 | (7) |
 | (8) |
 | (9) |
 are the Exradii (Johnson 1929, pp. 189-191).
As shown in Right Triangle, the inradius of a Right Triangle of integral side lengths  ,  , and  is alsointegral, and is given by
 | (10) |
 be the distance between inradius and Circumradius ,  . Then
 | (11) |
 | (12) |
Expressing the Midradius  and Circumradius in terms of the midradius gives
for an Archimedean Solid.See also Carnot's Theorem, Circumradius, Midradius
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. Mackay, J. S. ``Formulas Connected with the Radii of the Incircle and Excircles of a Triangle.'' Proc. Edinburgh Math. Soc. 12, 86-105. Mackay, J. S. ``Formulas Connected with the Radii of the Incircle and Excircles of a Triangle.'' Proc. Edinburgh Math. Soc. 13, 103-104.
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