Difference between revisions of "Exradius"
m (Some corrections to LaTeX, formatting and the formulas. Also added reference link.) |
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<math>r_1 = \frac{\Delta}{s-a} | <math>r_1 = \frac{\Delta}{s-a} | ||
= \sqrt{\frac{s(s-b)(s-c)}{s-a}} | = \sqrt{\frac{s(s-b)(s-c)}{s-a}} | ||
| − | = 4R\sin{\frac{ | + | = 4R\sin{\frac{A}{2}}\cos{\frac{B}{2}}\cos{\frac{C}{2}} |
</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> | ||
| + | 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 | ||
Latest revision as of 12:54, 21 January 2024
Excircle
The radius of an excircle. Let a triangle have exradius 
(sometimes denoted 
), opposite side of length 
and angle 
, area 
, and semiperimeter 
. Then
(Johnson 1929, p. 189), where 
is the circumradius. Let 
be the inradius, then
and
(Casey 1888, p. 65) and
Some fascinating formulas due to Feuerbach are
Reference:
Weisstein, Eric W. "Exradius." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Exradius.html
