If `(1)/(b-a)+ (1)/(b-c) = (1)/(a) +(1)/(c ) ` then prove that a, b,c are in H.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 the roots of the equation a(b-c)x^(2) + b(c-a) x+c(a-b)=0 are equal, then prove that a, b, c are in H.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 (b+c-a)/a,(c+a-b)/b,(a+b-c)/c are in A.P, prove that 1/a,1/b,1/c are also A.P.s

If the sum of roots of the equation (1)/(x+a)+(1)/(x+b)=(1)/(c ) is zero, prove that product of the roots is -(a^(2)+b^(2))/(2)

If a, b, c are in AP and a^(1/x)=b^(1/y)=c^(1/z) prove that x, y, z are in AP.

If b_(1)b_(2)= 2(c_(1)+c_(2)) , then prove that at least one of the equations x^(2) +b_(1)x+c_(1)=0 and x^(2) +b_(2)x+ c_(2)=0 has real roots.

If (b-c)^(2) , (c-a)^(2), (a-b)^(2) are in A.P. then prove that (1)/(b-c) , (1)/(c-a), (1)/(a-b) are also in A.P.

Suppose 1/(a+b),1/(2b),1/(b+c) are three consecutive terms of an AP. Prove that a,b,c are the consecutive terms of a G.P.