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

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

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):}) ?

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))]|

Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]| =x y z (x-y)(y-z)(z-x)(x+y+z)

Use properties of determinants to evaluate : |{:(x+y,y+z,z+x),(z,x,y),(1,1,1):}|

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