If A is the A.M. between a and b, prove that :
`(A-a)^(2) +(A-b)^(2)=1/2(a-b)^(2) `.

If A is the A.M. between a and b, prove that : 4(a-A)(A-b)=(a-b)^(2) .

If A_1 and A_2 are two A.M.’s between a and b, prove that : (2A_1-A_2)(2A_2-A_1) =ab .

If A_1 and A_2 are two A.M.’s between a and b, prove that : A_1+A_2 =a+b .

If a + b = 1, a gt 0, prove that (a + (1)/(a))^(2) + (b + (1)/(b))^(2) ge (25)/(2)

In any triangle ABC, prove that : (b^2-c^2)/(a^2)= sin (B -C)/sin(B+C) .

a, b, c are in G.P. and x and y are the A.M.’s between a, b and b, c respectively. Show that : 1/x+1/y=2/b .

In any triangle ABC, prove that : (tanC)/(tanA)= (b^2+c^2-a^2)/(a^2+b^2-c^2) .

a, b, c are in G.P. and x and y are the A.M.’s between a, b and b, c respectively. Show that : a/x+c/y=2 .

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

In any triangle ABC, prove that : (sin(B-C))/(sin(B+C))= (b^2 -c^2)/a^2 .