*Let a = first term of the AP.*

*and *

*Let d = common difference of the AP*

*Now*

*a = A+(p-1).d.......(1)*

*b = A+(q-1).d.......(2)*

*c = A+(r-1).d........(3)*

*Subtracting 2nd from 1st , 3rd from 2nd and 1st from 3rd we get*

*a-b = (p-q).d......(4)*

*b-c = (q-r).d........(5)*

*c-a = (r-p).d.......(6)*

*multiply 4,5,6 by c,a,b respectively we have*

*c.(a-b) = c.(p-q).d......(4)*

*a.(b-c) = a.(q-r).d........(5)*

*b.(c-a) = b.(r-p).d.......(6)*

*a(q-r).d+b(r-p).d+c(p-q).d = 0(a(q-r)+b(r-p)+c(p-q)).d = 0*

*Now since d is common difference it should be non zero*

*Hence*

*a(q-r)+b(r-p)+c(p-q)= 0*