Exradius

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)

(5) (Casey 1888, p. 65) and $rr_1r_2r_3=Delta^2.$ (6) Some fascinating formulas due to Feuerbach are $r(r_2r_3+r_3r_1+r_1r_2)=sDelta=r_1r_2r_3 r(r_1+r_2+r_3)=bc+ca+ab-s^2 rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=1/2(a^2+b^2+c^2)$

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Exradius