I have to prove if a,b,c <> 0 and
$\displaystyle \frac{{ay - bx}}{c} = \frac{{cx - az}}{b} = \frac{{bz - cy}}{a}$ then is $\displaystyle \frac{x}{a} = \frac{y}{b} = \frac{z}{c}$
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Write the common value $\displaystyle \frac{{ay - bx}}{c} = \frac{{cx - az}}{b} = \frac{{bz - cy}}{a} = t$ and treat it as a system of three linear equations in four unknowns x,y,z,t. Note that the rank of the system is three, since the determinant of the 3-by-3 minor corresponding to x,y,z is 2abc: hence there is a one-dimensional family of solutions. Since x=a, y=b, z=c, t=0 is clearly a solution, all solutions are multiples of this.