Let \(\newcommand{\vrp}{\varepsilon}\vrp_{AJ},\vrp_{JK},\vrp_{AK}\) be pairwise timelike events in a flat region, and \(\vrp_{AP}\) another event timelike and straight wrt. \(\vrp_{AJ},\vrp_{AK}\) and lightlike wrt. \(\vrp_{JK}\). Consequently, these four events are plane wrt. each other:
\[\newcommand{\sqv}{s^2[\vrp} 0=\begin{vmatrix}0&1&1&1&1\\1&0&\sqv_{AJ},\vrp_{AP}]&\sqv_{AJ},\vrp_{AK}]&\sqv_{AJ},\vrp_{JK}]\\1&\sqv_{AJ},\vrp_{AP}]&0&\sqv_{AJ},\vrp_{AK}]+\sqv_{AJ},\vrp_{AP}]-2\sqrt{\sqv_{AJ},\vrp_{AK}]\,\sqv_{AJ},\vrp_{AP}]}&0\\1&\sqv_{AJ},\vrp_{AK}]&0&\sqv_{AJ},\vrp_{AK}]+\sqv_{AJ},\vrp_{AP}]-2\sqrt{\sqv_{AJ},\vrp_{AK}]\,\sqv_{AJ},\vrp_{AP}]}&\sqv_{JK},\vrp_{AK}]\\1&\sqv_{AJ},\vrp_{JK}]&0&\sqv_{JK},\vrp_{AK}]&0\end{vmatrix}.\]
Thus:\[\frac{\sqv_{AJ},\vrp_{JK}]}{\sqv_{AJ},\vrp_{AK}]}=1-\sqrt{\frac{\sqv_{AJ},\vrp_{AP}]}{\sqv_{AJ},\vrp_{AK}]}}+\frac{\sqv_{JK},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AK}]}\left(1-\sqrt{\frac{\sqv_{AJ},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AP}]}}\right).\]

Now adding the following inequality:
\[2\sqrt{\frac{\sqv_{JK},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AK}]}}\le\frac{\sqv_{JK},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AK}]}\sqrt{\frac{\sqv_{AJ},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AJ}]}}+\sqrt{\frac{\sqv_{AJ},\vrp_{AP}]}{\sqv_{AJ},\vrp_{AK}]}}\]

and collecting yields:

\[\frac{\sqv_{AJ},\vrp_{JK}]}{\sqv_{AJ},\vrp_{AK}]}+2\sqrt{\frac{\sqv_{JK},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AK}]}}\le1+\frac{\sqv_{JK},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AK}]},\]

Therefore
\[
\sqrt{\frac{\sqv_{AJ},\vrp_{JK}]}{\sqv_{AJ},\vrp_{AK}]}}+\sqrt{\frac{\sqv_{JK},\vrp_{AK}]}{\sqv_{AJ},\vrp_{AK}]}}\le1,\]

a.k.a. reverse triangle inequality of (timelike) Lorentzian distances, https://en.wikipedia.org/wiki/Triangle_inequality#Reversal_in_Minkowski_space

#ProofInAToot #TwinFauxParadox #Relativity #Spacetime