Hence:

\[
   \underline{a}_{1}
   +t_1 \underline{a}_{12}
   =
   \underline{p}_{1}
   +\tau_1 \underline{p}_{12}
\]
\[
   \underline{a}_{1}
   +t_2 \underline{a}_{12}
   =
   \underline{p}_{1}
   +\tau_2 \underline{p}_{12}
\]

Thus:

\[
   (t_2-t_1) \underline{a}_{12}
   =
   (\tau_2-\tau_1) \underline{p}_{12}
\]