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Exradius

Revision as of 12:54, 21 January 2024 by Alexanderruan (talk | contribs)
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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 

$r_1	=	\frac{\Delta}{s-a}	 	=	\sqrt{\frac{s(s-b)(s-c)}{s-a}}	 	=	4R\sin{\frac{A}{2}}\cos{\frac{B}{2}}\cos{\frac{C}{2}}$

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

$4R=r_1+r_2+r_3-r$

and 

$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}$


(Casey 1888, p. 65) and 

$rr_1r_2r_3=\Delta^2$


Some fascinating formulas due to Feuerbach are 

$r(r_2r_3+r_3r_1+r_1r_2)=s\Delta=r_1r_2r_3$ 
$r(r_1+r_2+r_3)=bc+ca+ab-s^2$
$rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab$ 
$r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=\frac{1}{2}(a^2+b^2+c^2)$

Reference:

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

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