<s>[INST] Two circles tangent to each other and tangent to a line Two circles with centers $A$ and $B$ are externally tangent at point $C$.
 tangent to the two circles. Given that the radii of the two circles are $2cm$ and $3cm$, respectively, find $\frac{DC}{FC}$

 [/INST] Adjusting Matthew's figure:
$\hspace{2cm}$
Use Cosine theorem:
$$\frac{FC}{DC}=\frac{\sqrt{3^2+3^2-2\cdot 3^2\cdot \cos \beta}}{\sqrt{2^2+2^2-2\cdot 2^2\cdot \cos (180^\circ-\beta)}}=\sqrt{\frac{18-18\cdot \frac15}{8+8\cdot \frac15}}=\sqrt{\frac32}.$$
Note: $\cos \beta =\frac{BE}{AB}=\frac{FB-FE}{AC+BC}=\frac{FE-AD}{AC+BC}=\frac{3-2}{2+3}=\frac15$.
</s>