A line parallel to BC meets AB produced at D and AC produced at E in triangle ABC, so that B lies between A and D, and C lies between A and E. If AB = 6 cm, BD = 4 cm and AC = 9 cm, find CE (in cm).

BPT holds just as well when the sides are produced: the parallel line $DE$ cuts $AD$ and $AE$ proportionally, so $\frac{AB}{BD}=\frac{AC}{CE}$. Then $\frac{6}{4}=\frac{9}{CE}$ and $CE=\frac{9\times 4}{6}=6$ cm. (Check: $AD=10,AE=15$, and $\frac{6}{10}=\frac{9}{15}$.) The value 15 is $AE$, the whole produced side rather than the extra piece; 13.5 comes from inverting the ratio; 4 simply copies $BD$.