Question

If a, b, c are the sides of a right triangle where c is the hypotenuse, then prove that radius r of the circle touches the sides of the triangle is given by r = a+b+c/2

Answer 1

let the circle touches the sides  AB,BC and CA of triangle ABC at D, E and F 

as lengths of tangents drawn from an external point are equal  ...we have

 AD=AF,  BD=BE  and CE=CF

same as.... EB=BD=r

 then...  c = AF+FC

                   ∴ c = AD+CE

                   ⇒ c = (AB-DB)(CB-EB)

                   ⇒ c = a-r +b-r

                   ⇒ 2r =a+b-c

                       ⇒ r =( a+b-c)/2