Question

If a,b,c are the sides of a right angled triangle where c is the hypotenuse. Prove that the radius r of the circle which touches the sides of the triangle is given by r=a+b-c/2

Answer 1

The circle touches the sides BC, CA, AB of the right triangle ABC at D, E and F respectively.

Assume BC = a, CA = b and AB = c

Then AE = AF and BD = BF, CE = CD .

OE = OD = OF = r.

Here OEDC is a square then CE = CD = r.

i.e., b – r = AF, a – r = BF

or AB = c = AF + BF = b – r + a – r

∴ r = (a + b - c )/2