If `a ,b ,c` are in G.P., prove that: 1) `a(b^2+c^2)=c(a^2+b^2)` , 2) `A^2b^2c^2(1/(a^3)+1/(b^3)+1/(c^3))=a^3+b^3+c^3` , 3) `((a+b+c)^2)/(a^2+b^2+c^2)=(a+b+c)/(a-b+c)` , 4)`1/(a^2-b^2)+1/(b^2)=1/(b^2-c^2)` ,5) (a+2b=2c)(a-2b+2c)`=a^2+4c^2dot`

If a,b,c are in G.P.,prove that: a(b^(2)+c^(2))=c(a^(2)+b^(2))A^(2)b^(2)c^(2)((1)/(a^(3))+(1)/(b^(3))+(1)/(c^(3)))=a^(3)+b^(3)+c^(3)((a+b+c)^(2))/(a^(2)+b^(2)+c^(2))=(a+b+c)/(a-b+c)(1)/(a^(2)-b^(2))+(1)/(b^(2))=(1)/(b^(2)-c^(2))(a+2b=2c)(a-2b+2c)=a^(2)+4c^(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 .

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove that =|1 1 1 a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove that |{:(" "3,a+b+c,a^2+b^2+c^2),(a+b+c,a^2+b^2+c^2,a^3+b^3+c^3),(a^2+b^2+c^2,a^3+b^3+c^3,a^4+b^4+c^4):}|=|{:(1,1,1),(a,b,c),(a^2,b^2,c^2):}|^2

Prove that a^3+b^3+c^3-3abc=1/2(a+b+c){(a-b)^2+(b-c)^2+(c-a)^2}

Prove |{:(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ab,c^3):}|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

Prove: |[1,a^2+bc, a^3],[ 1,b^2+c a, b^3],[ 1,c^2+a b, c^3]|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)