If `A=2B`, then prove that either `c=b` or `a^2 = b(c+b)`

If a , b , c , are in A P ,a^2,b^2,c^2 are in HP, then prove that either a=b=c or a , b ,-c/2 from a GP (2003, 4M)

If the lines (a-b-c) x + 2ay + 2a = 0 , 2bx + ( b- c - a) y + 2b = 0 and (2c+1) x + 2cy + c - a - b = 0 are concurrent , then the prove that either a+b+ c = 0 or (a+b+c)^(2) + 2a = 0

If a ,b ,c are in A.P. and a^2, b^2, c^2 are in H.P., then prove that either a=b=cora ,b ,-c/2 form a G.P.

If the roots of the equation (c^2-a b)x^2-2(a^2-b c)x+b^2-a c=0 are equal, prove that either a=0 or a^3+b^3+c^3=3a b c dot

If a ,b ,c are in G.P. then prove that (a^2+a b+b^2)/(b c+c a+a b)=(b+a)/(c+b)

In DeltaABC, (b^(2)+c^(2))sin(B-C)=(b^(2)-c^(2))sin(B+C) , then prove that the triangle in either isosceles or right angled.

If (a-b),\ (b-c),(c-a) are in G.P. then prove that (a+b+c)^2=3(a b+b c+c a)

If in a triangle A B C , (2cosA)/a+(cos B)/b+(2cosC)/c=a/(b c)+b/(c a) , then prove that the triangle is right angled.

If in a triangle A B C , (2cosA)/a+(cos B)/b+(2cosC)/c=a/(b c)+b/(c a) , then prove that the triangle is right angled.

If a,b,c are in HP, then prove that (a+b)/(2a-b)+(c+b)/(2c-b)gt4 .