Proving a unique property in the projective space

Question

To prove that given three points $x,y,z\in{R}$ and a number $a$, there is only one $w\in{R}$ that satisfies: The proportion $[x,y;z,w]:=\frac{(x-z)(y-w)}{(x-w)(y-z)}$ equals to $a$. I don't know how it can exactly be showed. Is it right to take $\hat{w} \neq w$, suppose that it satisfies the above property and get $\hat{w} = w$ in contrast?

Answer 1

Say $$f(w)= \frac{(x-z)(y-w)}{(x-w)(y-z)}$$

We see that $f$ is fractional linear function (in general $f(w)={aw+b\over cw+d}$ which is injective if $ad-bc\ne 0$) and there for a conclusion.

Answer 2

Hint

Just solve the equation $$\frac{(x-z)(y-w)}{(x-w)(y-z)}=a$$ in w.

Answer 3

Say there exists $w'\ne w$ such that

$$\frac{(x-z)(y-w')}{(x-w')(y-z)}=\frac{(x-z)(y-w)}{(x-w)(y-z)}=a$$

then we have $$\frac{y-w'}{x-w'}=\frac{y-w}{x-w}$$

so $$(y-w')(x-w)= (x-w')(y-w)$$

so $$yx-w'x-wy+w'w = xy-wx-w'y+ww'\implies w'(y-x)= w(y-x)$$

and so $w=w'$.