It is a symbol that represents ratios; $A:B$ means "the ratio of $A$ to $B$". (Shows up all the time in Euclid; e.g., Book V.) When we write "$A:B=C:D$", we mean the ratio of $A$ to $B$ is the same as the ratio of $C$ to $D$.
Here he is talking about three ratios; the fact that $x:y:-1 = x_1:x_2:x_3$ means that the vectors determined by $(x,y,-1)$ and by $(x_1,x_2,x_3)$ are parallel. Note that $(x,y,-1) = (x,y,0) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and end point at $(x,y,0)$; and $(x_1,x_2,x_3-1) = (x_1,x_2,x_3) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and endpoint at $(x_1,x_2,x_3)$. Since $x:y:-1 = x_1:x_2:x_3$, they are parallel; since they both start at the same point, that means that the line through $(0,0,1)$ and $(x,y,0)$ (determined by the first vector) and the line through $(0,0,1)$ and $(x_1,x_2,x_3)$ (determined by the second vector) are the same, since they are parallel and they both go through $(0,0,1)$.
$[a,b]$ means the closed interval from $a$ to $b$, where $a$ and $b$ are elements of some ordered set, usually $\mathbb R$, in which case it's $\{x\in\mathbb R|a\le x\le b\}$. (But the notation is used also in $\mathbb R$-order trees, for example, and other sets.)
$f:[a,b]\to\mathbb R$ means $f$ is a function from $[a,b]$ to $\mathbb R$; it's continuous if its value at any point $c\in[a,b]$ is the limit of its values at $x\in[a,b]$ as $x\to c$.
Best Answer
It represents a ratio. Another way of writing it is $\frac a{a-b}=\frac c{c-d}$ and so on, with the conclusion $\frac ab = \frac cd$ and so on