If a,b,c are in G.P then show that `a(b^2-c^2)=c(a^2-b^2)`

If a,b,c,d are in G.P then show that (b-c)^2+(c-a)^2+(d-b)^2=(a-d)^2 .

If a,b,c are in G.P then show that a^2b^2c^2(1/a^3+1/b^3+1/c^3)=a^3+b^3+c^3 .

If a,b,c are in G.P then show that (a+2b+2c)(a-2b+2c)=a^2+4c^2

If a,b,c are in G.P., show that 1/(a+b),1/(2b),1/(b+c) are in A.P.

If a, b, c are in A.P. and a, b, c + 1 are in G.P. then prove that 4a=(a-c)^2 .

If a, b, c be the pth, qth and rth term of both an A.P. and G.P., then show that - a^(b-c).b^(c-a).c^(a-b)=1 .

If a + b + c = 0 , then prove that (2a^2)/(a^2-b^2-c^2)+(2b^2)/(b^2-c^2-a^2)+(2c^2)/(c^2-a^2-b^2)=3

If a+b+c=0 , then prove that (a^2+b^2+c^2)^2=4 (a^2b^2+b^2c^2+c^2a^2)

If a+b=c then prove that 1/(b^2+c^2-a^2)+1/(c^2+a^2-b^2)+1/(a^2+b^2-c^2)=0

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