Inradius

The radius of a Triangle's Incircle or of a Polyhedron's Insphere, denoted . For a Triangle,

			(1)
			(2)

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)

(4)

Other equations involving the inradius include

(5)

(6)

(7)

(8)

(9)

As shown in Right Triangle, the inradius of a Right Triangle of integral side lengths , , and is alsointegral, and is given by

(10)

(11)

(12)

Expressing the Midradius and Circumradius in terms of the midradius gives

			(13)
			(14)

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.

• Poisson-Charlier Function
• Poisson-Charlier Polynomial
• Poisson Distribution
• Poisson Integral
• Poisson Integral Representation
• Poisson Kernel
• Poisson Manifold
• Poisson's Bessel Function Formula
• Poisson's Equation
• Poisson's Harmonic Function Formula
• Poisson Sum Formula
• Poisson Trials
• Poke Move
• Poker
• Polar
• Polar Angle
• Polar Circle
• Polar Coordinates
• Polarity
• Polar Line
• Pole
• Pole (Geometry)
• Pole (Origin)
• Policeman on Point Duty Curve
• Polignac's Conjecture