Difference between revisions of "Exradius"
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Excircle | 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 | + | |
| − | <math> | + | 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 |
| − | r_1 = \Delta | + | |
| − | + | <math>r_1 = \frac{\Delta}{s-a} | |
| − | = sqrt | + | = \sqrt{\frac{s(s-b)(s-c)}{s-a}} |
| − | + | = 4R\sin{\frac{A}{2}}\cos{\frac{B}{2}}\cos{\frac{C}{2}} | |
| − | = | ||
| − | |||
</math> | </math> | ||
| − | (Johnson 1929, p. 189), where R is the circumradius. Let r be the inradius, then | + | (Johnson 1929, p. 189), where <math>R</math> is the circumradius. Let <math>r</math> be the inradius, then |
| − | <math> | + | |
| − | + | <math>4R=r_1+r_2+r_3-r</math> | |
| − | </math> | + | |
| − | + | and | |
| − | <math> | + | |
| − | 1 | + | <math>\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{r}</math> |
| − | </math> | + | |
| − | + | ||
(Casey 1888, p. 65) and | (Casey 1888, p. 65) and | ||
| − | <math> | + | |
| − | + | <math>rr_1r_2r_3=\Delta^2</math> | |
| − | </math> | + | |
| − | + | ||
Some fascinating formulas due to Feuerbach are | Some fascinating formulas due to Feuerbach are | ||
| − | <math> | + | |
| − | + | <math>r(r_2r_3+r_3r_1+r_1r_2)=s\Delta=r_1r_2r_3</math> | |
| − | r(r_1+r_2+r_3)=bc+ca+ab-s^2 | + | <math>r(r_1+r_2+r_3)=bc+ca+ab-s^2 </math> |
| − | rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab | + | <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=1 | + | <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> |
| − | </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 
(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
