Question

If r is the radius of incircle of $\Delta$ ABC, prove the following formula :
r = S/s

Answer 1

The bisectors of the angles of $\Delta$ ABC meet in I. Draw ID, IE, IF perpendicular to the sides from I.
By Geometry, $ID=IE=IF=r$
$\Delta$ ABC = $\Delta$ IBC + $\Delta$ ICA + $\Delta$ IAB
or $S=\frac{1}{2}ar+\frac{1}{2}br+\frac{1}{2}cr$
or $S=\frac{1}{2}r(a+b+c)=\frac{1}{2}r.$ 2s = r . s
$\therefore$ r = S/s