In ∆ A B C , suppose the radius of the circle opposite to an angle A is denoted by r 1 , similarly r 2 ↔ angle B , r 3 ↔ angle C . If ' r ' is the radius of inscribed circle then, what is the value of a b - r 1 r 2 r 3 =

Step-by-step solution

We know, r 1 = ∆ s - a ,   r 2 = ∆ s - b ,   r 3 = ∆ s - c Then, a b - r 1 r 2 = a b - ∆ 2 s - a s - b = a b - s s - a s - b s - c s - a s - b = a b - s s - c = a b - a + b + c 2 a + b - c 2 = 4 a b - a + b 2 + c 2 4 = c 2 - a - b 2 4 = c + a - b 2 × c - a + b 2 = s - b s - a Now, a b - r 1 r 2 r 3 = s - b s - a ∆ s - c = s s - a s - b s - c s ∆ = ∆ s = r