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Exradius: Difference between revisions

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


<math>r_1 = \frac{\Delta}{(s-a)}
<math>r_1 = \frac{\Delta}{s-a}
(1)
= \sqrt{\frac{s(s-b)(s-c)}{s-a}}
= \sqrt{\frac{(s(s-b)(s-c))}{(s-a)}}
= 4R\sin{\frac{A}{2}}\cos{\frac{B}{2}}\cos{\frac{C}{2}}
(2)
= 4R\sin{\frac{1}{2A}}\cos{\frac{1}{2B}}\cos{\frac{1}{2C}}
(3)
</math>
</math>
(Johnson 1929, p. 189), where <math>R</math> is the circumradius. Let <math>r</math> be the inradius, then
(Johnson 1929, p. 189), where <math>R</math> is the circumradius. Let <math>r</math> be the inradius, then
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  <math>4R=r_1+r_2+r_3-r</math>
  <math>4R=r_1+r_2+r_3-r</math>


(4)
and
 
<math>\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}</math>


<math>\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=1/r</math>


(5)
(Casey 1888, p. 65) and
(Casey 1888, p. 65) and


  <math>rr_1r_2r_3=\Delta^2</math>
  <math>rr_1r_2r_3=\Delta^2</math>


(6)
 
Some fascinating formulas due to Feuerbach are
Some fascinating formulas due to Feuerbach are


  <math>r(r_2r_3+r_3r_1+r_1r_2)=s\Delta=r_1r_2r_3</math>  
  <math>r(r_2r_3+r_3r_1+r_1r_2)=s\Delta=r_1r_2r_3</math>  
<math>r(r_1+r_2+r_3)=bc+ca+ab-s^2 </math>
<math>r(r_1+r_2+r_3)=bc+ca+ab-s^2 </math>
<math>rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab</math>  
<math>rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab</math>  
$r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=\frac{1}{2(a^2+b^2+c^2)}
<math>r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=\frac{1}{2}(a^2+b^2+c^2)</math>
 
Reference:
 
Weisstein, Eric W. "Exradius." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Exradius.html

Latest revision as of 12:54, 21 January 2024

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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