GS. 1 In the figure, there are two common tangents. These common tangents meet the circles at points $A, B, C$ and $D$. If $A C=9 \mathrm{~cm}$, $B D=3 \mathrm{~cm}, \angle B A C=60^{\circ}$ and $A B=s \mathrm{~cm}$, find the value of $s$.
Produce $A B$ and $C D$ to meet at $E$.
$$
A E=C E, B E=D E \text { (tangent from ext. pt.) }
$$
$\triangle E A C$ and $\triangle E B D$ are isosceles triangles
$$
\begin{array}{l}
\angle E C A=\angle B A C=60^{\circ} \text { (base } \angle \text { s isos. } \triangle \text { ) } \\
\angle A E C=60^{\circ}(\angle \mathrm{s} \text { sum of } \triangle) \\
\angle E B D=\angle E D B=60^{\circ}(\angle \text { s sum of } \triangle, \text { base } \angle \text { s isos. } \triangle \text { ) }
\end{array}
$$
$\therefore \triangle E A C$ and $\triangle E B D$ are equilateral triangles
$$
\begin{array}{l}
E B=B D=3 \mathrm{~cm}, E A=A C=9 \mathrm{~cm} \text { (sides of equilateral triangles) } \\
s=9-3=6
\end{array}
$$