Derivation of Formula for Radius of Incircle
The radius of incircle is given by the formula
r=Ats
where At = area of the triangle and s = semi-perimeter.
Derivation
Let
At = Area of triangle ABC
At = Area of triangle BOC + Area of triangle AOC + Area of triangle AOB
At=ABOC+AAOC+AAOB
At=12ar+12br+12cr
At=12(a+b+c)r
Let 12(a+b+c)=s, the semi-perimeter
Thus,
At=sr
r=Ats