Exradius: Difference between revisions
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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 | 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> | <math> | ||
r_1 = Delta/(s-a) | r_1 = \Delta/(s-a) | ||
(1) | (1) | ||
= sqrt((s(s-b)(s-c))/(s-a)) | = sqrt((s(s-b)(s-c))/(s-a)) | ||
| Line 10: | Line 10: | ||
</math> | </math> | ||
(Johnson 1929, p. 189), where R is the circumradius. Let r be the inradius, then | (Johnson 1929, p. 189), where R is the circumradius. Let r be the inradius, then | ||
<math> | |||
4R=r_1+r_2+r_3-r | 4R=r_1+r_2+r_3-r | ||
</math> | |||
(4) | (4) | ||
1/(r_1)+1/(r_2)+1/(r_3)=1/r | <math> | ||
1/(r_1)+1/(r_2)+1/(r_3)=1/r | |||
</math> | |||
(5) | (5) | ||
(Casey 1888, p. 65) and | (Casey 1888, p. 65) and | ||
<math> | |||
rr_1r_2r_3=Delta^2. | rr_1r_2r_3=Delta^2. | ||
</math> | |||
(6) | (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)=sDelta=r_1r_2r_3 | r(r_2r_3+r_3r_1+r_1r_2)=sDelta=r_1r_2r_3 | ||
r(r_1+r_2+r_3)=bc+ca+ab-s^2 | 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 | 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=1/2(a^2+b^2+c^2) | r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=1/2(a^2+b^2+c^2) | ||
</math> | |||
Revision as of 21:18, 26 June 2019
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
(Johnson 1929, p. 189), where R is the circumradius. Let r be the inradius, then
(4)
(5)
(Casey 1888, p. 65) and
(6)
Some fascinating formulas due to Feuerbach are
