Determinant form of `(x-y)(y-z)(z-x)`

Determinant , form (x-y)(y-z)(z-x)(xy+yz+zx), of

Determinant (x-y)(y-z)(z-x)(x+y+z)

The simplified form of x(y-z)(y+z)+y(z-x)(z+x)+z(x-y)(x+y) is

Area of the triangle formed by (x,y+z),(y,z+x),(z,x+y) is

The repeated factor of the determinant |(y +z,x,y),(z +x,z,x),(x +y,y,z)| , is

What is the determinant of the matrix ({:(x,y,y+z),(z,x,z+x),(y,z,x+y):}) ?

Find the value of the "determinant" |{:(1,x,y+z),(1,y,z+x),(1,z,x+y):}|

Without actual expansion, prove that the following determinants vanish : |{:(x+y,y+z,z+x),(z,x,y),(1,1,1):}|

The value of the determinant ,log_(a)((x)/(y)),log_(a)((y)/(z)),log_(a)((z)/(x))log_(b)((y)/(z)),log_(b)((z)/(x)),log_(b)((x)/(y))log_(c)((z)/(x)),log_(c)((x)/(y)),log_(c)((y)/(z))]|

Using properties of determinants, prove that |[2y,y-z-x,2y],[2z,2z, z-x-y],[ x-y-z, 2x,2x]|=(x+y+z)^3