If a,b,c,x`in`R and `(x-a+b)^2 + (x-b+c)^2 = 0`, prove that a,b,c are in A.P.

If a,b,c are in A.P. then :

If the roots of the equation : a(b - c)x^2 +b (c - a)x + c(a - b) = 0 are equal, show that 1/a,1/b,1/c are in A.P.

If a(1/b+1/c), b(1/c+1/a), c(1/a+1/b) are in A.P, prove that a, b, c are in A.P.

If a(1/b +1/c) , b(1/c +1/a), c(1/a + 1/b) are in A.P. Prove that a,b,c are in A.P.

If in Delta ABC, (a -b) (s-c) = (b -c) (s-a) , prove that r_(1), r_(2), r_(3) are in A.P.

If 1/a+1/(a-b)+1/c+1/(c-b)= 0 and a + c-b!=0 , then prove that a, b, c are in H.P.

If a+b+c ne 0 and (b + c)/a, (c + a)/b, (a + b)/c are in A.P., prove that : 1/a,1/b,1/c are also in A.P.

If a, b, c are in A.P., prove that : b+c,c+a,a+ b are also in A.P.

In a triangle ABC , if (sinA)/(sinC)=(sin(A-B))/(sin(B-C)) . Prove that a^2,b^2,c^2 are in A.P.

If a, b, c are in A.P, b, c, d are in G.P. and 1/c,1/d,1/e are in A.P. prove that a, c, e are in G.P.