special relativity - How to express in terms of spacetime intervals the average speed of one participant having moved between two suitable others

Given a flat region of spacetime as set $\mathcal S$ of events together with values of spacetime intervals (up to a common non-zero constant) for each pair of events, $s^2 : \mathcal S \times \mathcal S \rightarrow \mathbb R$,
and considering three (distinct) participants $A$, $B$, and $P$ contained in this region, such that

• $A$ and $B$ had been and remained at rest to each other,


• $A$ and $P$ had been passing each other (i.e. as coincidence event $\varepsilon_{AP} \in \mathcal S)$, and



• $B$ and $P$ had been passing each other (i.e. as coincidence event $\varepsilon_{BP} \in \mathcal S$),

how can $\| \overline{ \mathbf v}_{AB}[~P~] \|$, i.e. the average speed of $P$'s motion with respect to $A$ and $B$, be expressed in terms of the given spacetime interval values ?