If (b - c)², (c - a)², (a - b)² are in A.P.,

then  will be in

Concept:

In A.P common difference is same. i.e. If a, b, c are in A.P, then

b - a = c - b

Calculation:

Given:

(b - c)2, (c - a)2, (a - b)2 are in A.P

⇒ (c - a)2 - (b - c)2 = (a - b)2 - (c - a)2 

(c - a + b - c)(c - a - b + c) = (a - b + c - a)(a - b - c + a)

(b - a)(2c - a - b) = (c - b)(2a - b - c)   ...(i)

Now, for  

(b + a - 2c)(a - b) = (c + b - 2a)(b - c)

(b - a)(2c - a - b) = (c - b)(2a - b - c)   ...(ii)

As equation (i) & (ii) are equal,

Hence  is in A.P