Question

prove with explanation.

Answer 1

if S denotes the area of the triangle ABC, then S = area of triangles AIC+BIC+AIB but area of triangleAIC=(1/2)r*b where b denotes the length of side AC. similarly, area of triangleBIC=(1/2)r*a and area of triangleAIB=(1/2)r*c where a and c denote the lengths of sides BC and AB respectively. It is easy to note that the inradius is perpendicular to the sides as they are tangents to the in circle.
Thus adding the above results we get,
S = (1/2)r*(a+b+c) = r*(a+b+c)/2 = r*s where s denotes the semiperimeter of
triangle ABC which equals (a+b+c)/2.
so, r= area of triangle/s