Show that `a^(2)(1+b^(2))+b^(2)(1+c^(2))+c^(2)(1+a^(2))gt abc`

In triangle ABC if A:B:C=1:2:4," then " (a^(2)-b^(2))(b^(2)-c^(2))(c^(2)-a^(2))=lambda a^(2)b^(2)c^(2) , where lambda= (where notations have their usual meaning)

In triangle ABC if A:B:C=1:2:4," then " (a^(2)-b^(2))(b^(2)-c^(2))(c^(2)-a^(2))=lambda a^(2)b^(2)c^(2) , where lambda= (where notations have their usual meaning)

Using properties of determinant, Show that |{:((b+c)^(2),a^(2),a^(2)),(b^(2),(c+a)^(2),b^(2)),(c^(2),c^(2),(a+b)^(2)):}|=2abc(a+b+c)^(3)

If a^(2) + b^(2) + c^(3) + ab + bc + ca le 0 for all, a, b, c in R , then the value of the determinant |((a + b +2)^(2),a^(2) + b^(2),1),(1,(b +c + 2)^(2),b^(2) + c^(2)),(c^(2) + a^(2),1,(c +a +2)^(2))| , is equal to

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

If a, b, c are in geometric progression, then prove that : (1)/(a^(2)-b^(2))+(1)/(b^(2))=(1)/(b^(2)-c^(2))

If a+ bi = (c + i)/(c-i), a, b , c in R , show that a^(2) + b^(2)=1 and (b)/(a)= (2c)/(c^(2)-1)

If a, b, c are three distinct positive real numbers, then the least value of ((1+a+a^(2))(1+b+b^(2))(1+c+c^(2)))/(abc) , is

If a,b, and c are all different and if |{:(a,a^(2),1+a^(3)),(b,b^(2),1+b^(3)),(c,c^(2),1+c^(3)):}| =0 Prove that abc =-1.

Prove that [[1+a^2+a^4, 1+a b+a^2b^2 ,1+a c+a^2c^2],[ 1+a b+a^2b^2, 1+b^2+b^4, 1+b c+b^2c^2],[ 1+a c+a^2c^2, 1+b c+b^2c^2, 1+c^2+c^2]]=(a-b)^2(b-c)^2(c-a)^2