Solution:

Denote the vertices of the heptagon by $A$, $B$, $C$, $D$, $E$, $F$, $G$. Since an alternating arrangement cannot be continued all the way around the heptagon, two adjacent vertices must be the same color, say $A$ and $B$. If any of $C$, $E$, $G$ shares this color, we are done since triangles $A B C$, $A B E$, and $A B G$ are all isosceles. On the other hand, if $C$, $E$, and $G$ are all of the opposite color, we are also done because triangle $C E G$ is isosceles. Thus in all cases we can find an isosceles triangle.