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Exradius

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

The Radius of an Excircle. Let a Triangle have exradius $r_1$ (sometimes denoted $\rho_1$), opposite side $a_1$ and angle $\alpha_1$, Area $\Delta$, and Semiperimeter $s$. 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 $R$ is the Circumradius. Let $r$ 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.



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© 1996-9 Eric W. Weisstein
1999-05-25