If the roots of the equation  (b-c)x2 +(c-a)x +(a-b) = 0  are equal  than prove that 

2b = c + a  

here is your solution

(b-c)x² +(c-a)x +(a-b) = 0

if the roots of this equation in equal than D = o i.e,

D = 0

means..

b² - 4ac = 0

bere b = (c-a)

         a = (b-c)

         c = (a-b)

by putting these values in equation 1... we get...

(c-a)² - 4 (b-c) (a-b) = 0

c²+a²-2ac - 4(ab-b²-ac+bc) = 0

c²+a²-2ac-4ab+4b²+4ac-4bc = 0

a²+4b²+c²-4ab-4bc+2ac = 0

(a-2b+c)² = 0

so....

a-2b+c = 0

a+c = 2b