Expand the following and verify
(i) ` (a+b+c)^(2) =(-a-b-c)^(2) `
(ii) ` (-a+b+c)^(2) =(a-b-c)^(2) `
(iii) ` (a-b-c)^(2) =(-a+b-c)^(2) `
(iv) ` (a+b-c)^(2) =(-a-b+c)^(2) `

Expand : (a +b -c)^(2)

Show that |((a+b)^(2),(a-b)^(2),ab),((b+c)^(2),(b-c)^(2),bc),((c+a)^(2),(c-a)^(2),ca)|

In a Delta ABC, prove that (a sin(B - C))/(b^(2) - c^(2)) = (b sin (C - A))/(c^(2) - a^(2)) = (c sin (A - B))/(a^(2) - b^(2))

Without expanding the determinant, prove that {:[( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) ]:} ={:[( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) ]:}

If a,b,c,d are in geometric sequence then prove that (b-c)^(2) +(c-a)^(2) -(d-b)^(2)=(a-d)^2

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

By using properties of determinants , show that : (i) {:[( 1,a,a^(2)),( 1,b,b^(2)),( 1,c,c^(2))]:}=(a-b)(b-c) (c-a) (ii) {:[( 1,1,1),( a,b,c) ,(a^(3) , b^(3), c^(3))]:} =( a-b) (b-c)( c-a) (a+b+c)

With usual notations, in triangle A B C ,acos(B-C)+bcos(C-A)+c"cos"(A-B) is equal to(a) (a b c)/R^2 (b) (a b c)/(4R^2) (c) (4a b c)/(R^2) (d) (a b c)/(2R^2)