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)

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)