Exradius

$\begin{figure}\begin{center}\BoxedEPSF{Excircle.epsf}\end{center}\end{figure}$

The Radius of an Excircle. Let a Triangle have exradius (sometimes denoted $\rho_1$ ), opposite side and angle $\alpha_1$ , Area $\Delta$ , and Semiperimeter . Then

$\displaystyle {r_1}^2$	$\textstyle =$	$\displaystyle \left({\Delta\over s-a_1}\right)^2$	(1)
	$\textstyle =$	$\displaystyle {s(s-a_2)(s-a_3)\over s-a_1}$	(2)
	$\textstyle =$	$\displaystyle 4R\sin({\textstyle{1\over 2}}\alpha_1)\cos({\textstyle{1\over 2}}\alpha_2)\cos({\textstyle{1\over 2}}\alpha_3)$	(3)

(Johnson 1929, p. 189), where

is the Circumradius. Let

be the Inradius, then

$\begin{displaymath} 4R=r_1+r_2+r_3-r \end{displaymath}$

(4)

$\begin{displaymath} {1\over r_1}+{1\over r_2}+{1\over r_3}={1\over r} \end{displaymath}$

(5)

$\begin{displaymath} r r_1 r_2 r_3=\Delta^2. \end{displaymath}$

(6)

Some fascinating Formulas due to Feuerbach are

$\begin{displaymath} r_2r_3+r_3r_1+r_1r_3=s^2 \end{displaymath}$

(7)

$\begin{displaymath} r(r_2r_3+r_3r_1+r_1r_2)=s\Delta=r_1r_2r_3 \end{displaymath}$

(8)

$\begin{displaymath} r(r_1+r_2+r_3)=a_2a_3+a_3a_1+a_1a_2-s^2 \end{displaymath}$

(9)

$\begin{displaymath} rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=a_2a_3+a_3a_1+a_1a_2 \end{displaymath}$

(10)

$\begin{displaymath} r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3-{\textstyle{1\over 2}}({a_1}^2+{a_2}^2+{a_3}^2) \end{displaymath}$

(11)

(Johnson 1929, pp. 190-191).

See also Circle, Circumradius, Excircle, Inradius, Radius

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. ``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.