The inradius r is the radius of the inscribed circle (incircle), tangent to all three sides. r = $\frac{Delta}{s}$, where s is the semi-perimeter. Alternative forms: r = (s-a)tan$\frac{A}{2}$ = (s-b)tan$\frac{B}{2}$ = (s-c)tan$\frac{C}{2}$, and r = 4R sin$\frac{A}{2}$sin$\frac{B}{2}$sin$\frac{C}{2}$. The three exradii r1, r2, r3 are radii of excircles opposite to vertices A, B, C respectively. r1 = $\frac{Delta}{s-a}$ = stan$\frac{A}{2}$ = 4R sin$\frac{A}{2}$cos$\frac{B}{2}$cos$\frac{C}{2}$, and similarly for r2, r3. Key relationships: r1 + r2 + r3 - r = 4R, r1r2r3 = r$s^2$, 1/r = 1/r1 + 1/r2 + 1/r3. The Euler relation connecting circumcenter O and incenter I: $OI^2$ = $R^2$ - 2Rr = R(R-2r), which also proves r <= R/2 (Euler's inequality).