KöMaL 2016 A676

Let $\mathcal{K}$ be a circle with diameter $OI$. Construct a one-to-one correspondence $f \colon \mathcal{K}\setminus \{O,I\}\to\mathbb{R}\setminus\{0\}$ with the following property: for every four distinct points $A$, $B$, $C$, $D$ of $\mathcal{K}$, other than $O$ and $I$, $f(A)f(B)=f(C)f(D)$ is satisfied if and only if the lines $AB$, $CD$ and $OI$ are concurrent or parallel to each other.