(c) If `(a)/(b)=(b)/(c)` ,prove that `(a+b+c)(a-b+c)=a^(2)+b^(2)+c^(2)`

if a:b = b:c prove that ((a+b)/(b+c))^2 = (a^2+b^2)/(b^2+c^2) .

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

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

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

Prove that ((a+b+c)(b+c-a)(c+a-b)(a+b-c))/(4b^(2)c^(2))=sin^(2)A

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

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

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

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

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