If `x+y+2z=0` then prove that `x^2/(yz)+y^2/(zx)+(8z^2)/(xy)=6`

If x + y + z = 0 , then the value of (x^2)/(yz) + (y^2)/(zx) + (z^2)/(xy) is:

If x^2/(yz)+y^2/(zx)+z^2/(xy)=3 then (x+y+z)^3=

If x prop y and y prop z , then prove that : (x^2+y^2+z^2) prop (xy+yz+zx) .

If xy + yz + zx = 1 , prove that : x/(1-x^2)+y/(1-y^2)+z/(1-z^2)= (4xyz)/((1-x^2)(1-y^2)(1-z^2)) .

If x,y and z are three different numbers, then prove that : x^(2) + y^(2) + z^(2)- xy - yz- zx is always positive

Prove that |[1,x,x^2-yz],[1,y,y^2-zx],[1,z,z^2-xy]|= 0

By using properties of determinants, prove that |[x,x^2,yz],[y,y^2,zx],[z,z^2,xy]|=(x-y)(y-z)(z-x)(xy+yz+zx)