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

$r_1 = \frac{\Delta}{(s-a)} (1) = \sqrt{\frac{(s(s-b)(s-c))}{(s-a)}} (2) = 4R\sin{\frac{1}{2A}}\cos{\frac{1}{2B}}\cos{\frac{1}{2C}} (3)$ (Johnson 1929, p. 189), where $R$ is the circumradius. Let $r$ be the inradius, then

(4)

$\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=1/r$

(5) (Casey 1888, p. 65) and

(6) Some fascinating formulas due to Feuerbach are

$r(r_1+r_2+r_3)=bc+ca+ab-s^2$ $rr_1+rr_2+rr_3+r_1r_2+r_2r_3+r_3r_1=bc+ca+ab$ $r_2r_3+r_3r_1+r_1r_2-rr_1-rr_2-rr_3=\frac{1}{2(a^2+b^2+c^2)}

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Exradius