Solution:
Let $A, B, C, D$, and $E$ be the known vertices and $F$ and $G$ the unknown vertices. The arrangement of $A, B, C, D$, and $E$ depends on the relative positions of $F$ and $G$ as shown in the diagram.

By connecting the intersections of $B C$ with $D E$ and $B D$ with $C E$ one finds $l$, the line of symmetry through $G$. Let $A B$ and $A C$ meet $l$ in $H$ and $I$ respectively. By symmetry about $l, H E$ and $I D$ both pass through $F$. Now perform the same construction on $B, C, D, E, F$ to get $G$.