Question

Eliminate $x,y,z$ from the equations $(x+y-z)(x-y+z)=ayz,(y+z-x)(y-z+x)=bzx,(z+x-y)(z-x+y)=cxy$.

Answer 1

By multiplication

$abc{x}^2{y}^2{z}^2={(y+z-x)}^2{(z+x-y)}^2{(x+y-z)}^2$

$=(-x^3-y^3-z^3+y^{2}z+y{z}^2+z^{2}x+z{x}^2+x^{2}y+x{y}^2-2xyz)$

Then $abc={(\frac{y}{z}+\frac{z}{y}+\frac{z}{x}+\frac{x}{z}+\frac{x}{y}+\frac{y}{x}-\frac{x^2}{yz}-\frac{y^2}{zx}-\frac{z^2}{xy}-2)}^2$

$\therefore abc={((2-a)+(2-b)+(2-c)-2)}^2$

$abc={(4-a-b-c)}^2$