Consider the extremely simple function $$f(a,b)=\frac a{a-b}.$$ This gives the coordinate where the line through $(0,a)$ and $(1,b)$ meets the $x$-axis.
I noticed that the function $f$ has some interesting properties: $$\begin{gather} f(a,b)+f(b,a)=1, \\ f(a,x)f(a,y) = f(a,x)f(x,y) + f(a,y)f(y,x), \\ f(a,x)f(b,x) = f(a,b)f(b,x) + f(b,a)f(a,x), \end{gather}$$ These identities can be verified by calculation, but for the latter two it is hard to see a priori that they should be true.
By the way, if you imagine $f(a,b)$ as a directed edge from $a$ to $b$ in a graph, the three identities have an elegant graphical representation: $$\begin{gather} {}\rightarrow{} + {}\leftarrow{} = 1 \\[1em] {}\nwarrow\nearrow{} = {}\overrightarrow{\nwarrow\ \ }{} + {}\overleftarrow{\ \ \nearrow} \\[1em] {}\searrow\swarrow{} = {}\overrightarrow{\ \ \swarrow}{} + {}\overleftarrow{\searrow\ \ } \\[1em] \end{gather}$$
I have a couple of questions:
Does $f$ fall into any well-studied class of functions? It's not even commutative or associative, but those properties above are interesting.
Is there an elegant way to prove these identities without laborious calculation, or to discover them (and any others I might have missed) starting only from the definition of $f$?
$f(a,b)$ is the cross-ratio $(a,\infty;0,b)$.