Convex Combination of $\underline{p}_1, \underline{p}_2$:
\[
   \underline{x}
   =
   \underline{p}_1+\tau \underline{p}_{12}
   =
   \underline{p}_2+\tau' \underline{p}_{21},
   \quad \tau=1-\tau' \in \mathbb{R}
\]
$\underline{p}_{12}=\underline{p}_{2}-\underline{p}_{1}$.

Particularly, for $\underline{a}_1$:

\[
   \underline{a}_1
   =
   \underline{p}_1+\tau_1 \underline{p}_{12}
   =
   \underline{p}_2+\tau_1' \underline{p}_{21}
\]

And $\underline{a}_2$:

\[
   \underline{a}_2
   =
   \underline{p}_1+\tau_2 \underline{p}_{12}
   =
   \underline{p}_2+\tau_2'\underline{p}_{21}
\]