Difference between revisions of "Exradius"
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Excircle | Excircle | ||
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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}{ | + | <math>r_1 = \frac{\Delta}{s-a} |
− | + | = \sqrt{\frac{s(s-b)(s-c)}{s-a}} | |
− | = \sqrt{\frac{ | ||
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= 4R\sin{\frac{1}{2A}}\cos{\frac{1}{2B}}\cos{\frac{1}{2C}} | = 4R\sin{\frac{1}{2A}}\cos{\frac{1}{2B}}\cos{\frac{1}{2C}} | ||
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</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> | ||
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− | + | <math>\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=1/r</math> | |
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(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> | ||
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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> |
− | + | <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> | |
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+ | Reference: | ||
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+ | Weisstein, Eric W. "Exradius." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Exradius.html |
Revision as of 21:04, 25 July 2023
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
(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