If `x/(b-c)=y/(c-a)=z/(a-b)`, prove that `ax+by+cz=0`

If x=b-c+a, y=c-a+b, z=a-b+c , then prove that (b-a) x + (c-b)y +(a-c)z=0

a/(y-z) = b/(z-x) = c/(x-y) then ax + by + cz =___

If x^2/(by+cz)=y^2/(cz+ax)=z^2/(ax+by)=1 then show that a/(a+x)+b/(b+y)+c/(c+z)=1

If (by+cz)/(b^2+c^2)=(cz+ax)/(c^2+a^2)=(ax+by)/(a^2+b^2) then prove that x/y=y/b=z/c

If (x+y)/(3a-b) = (y+z)/(3b-c) = (z+x)/(3c-a) , then show that (x+y+z)/(a+b+c)= (ax + by+cz)/(a^(2)+b^(2)+c^(2))

If (x^2-yz)/a=(y^2-zx)/b=(z^2-xy)/c , show that (a+b+c)(x+y+z)=(ax+by+cz).

If x+y+z=0 prove that |a x b y c z c y a z b x b z c x a y|=x y z|a b cc a bb c a|

If ax^(10)=by^(10)=cz^(10) and (1)/(x)+(1)/(y)+(1)/(z)=1 then prove that (ax^(9)+by^(9)+cz^(9))^(1/10)=a^(1/10)+b^(1/10)+c^(1/10)

If x^(2) : (by + cz) =y^(2) : (cz + ax) = z^(2) : (ax + by) = 1 , then show that a/(a+x) + b/(b+y) + c/(c+z) = 1 .