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Exradius

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Excircle

The radius of an excircle. Let a triangle have exradius r_A (sometimes denoted rho_A), opposite side of length a and angle A, area Delta, and semiperimeter s. Then

r_1=Delta/(s-a)
(1)
=sqrt((s(s-b)(s-c))/(s-a))
(2)
=4Rsin(1/2A)cos(1/2B)cos(1/2C)
(3)

(Johnson 1929, p. 189), where R is the circumradius. Let r be the inradius, then

4R=r_1+r_2+r_3-r
(4)
1/(r_1)+1/(r_2)+1/(r_3)=1/r
(5)

(Casey 1888, p. 65) and

rr_1r_2r_3=Delta^2.
(6)

Some fascinating formulas due to Feuerbach are

r(r_2r_3+r_3r_1+r_1r_2)=sDelta=r_1r_2r_3 r(r_1+r_2+r_3)=bc+ca+ab-s^2 rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=1/2(a^2+b^2+c^2)
(7)

(Johnson 1929, pp. 190-191).

See also

Circle, Circumradius, Excircles, Inradius, Radius

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.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. "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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Exradius

Cite this as:

Weisstein, Eric W. "Exradius." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Exradius.html

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