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 | 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 | ||
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| − | r_1 = | + | r_1 = Delta/(s-a) |
(1) | (1) | ||
= sqrt((s(s-b)(s-c))/(s-a)) | = sqrt((s(s-b)(s-c))/(s-a)) | ||
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= 4Rsin(1/2A)cos(1/2B)cos(1/2C) | = 4Rsin(1/2A)cos(1/2B)cos(1/2C) | ||
(3) | (3) | ||
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(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 | ||
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4R=r_1+r_2+r_3-r | 4R=r_1+r_2+r_3-r | ||
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(4) | (4) | ||
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1/(r_1)+1/(r_2)+1/(r_3)=1/r | 1/(r_1)+1/(r_2)+1/(r_3)=1/r | ||
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(5) | (5) | ||
(Casey 1888, p. 65) and | (Casey 1888, p. 65) and | ||
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rr_1r_2r_3=Delta^2. | rr_1r_2r_3=Delta^2. | ||
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(6) | (6) | ||
Some fascinating formulas due to Feuerbach are | Some fascinating formulas due to Feuerbach are | ||
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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) | ||
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Revision as of 21:19, 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
r_1 = Delta/(s-a) (1) = sqrt((s(s-b)(s-c))/(s-a)) (2) = 4Rsin(1/2A)cos(1/2B)cos(1/2C) (3)
(Johnson 1929, p. 189), where R is the circumradius. Let r be the inradius, then
4R=r_1+r_2+r_3-r
(4)
1/(r_1)+1/(r_2)+1/(r_3)=1/r
(5) (Casey 1888, p. 65) and
rr_1r_2r_3=Delta^2.
(6) Some fascinating formulas due to Feuerbach are
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 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)
