A student writes: "In triangle ABC, DE is parallel to BC with D on AB and E on AC; AD = 5 cm, DB = 8 cm, AE = 10 cm and AC = 18 cm." Why is this set of measurements impossible?

BPT requires $\frac{AD}{DB}=\frac{AE}{EC}$, i.e. $\frac{5}{8}=\frac{10}{EC}$, so $EC$ must be 16 cm and $AC=AE+EC=26$ cm. The quoted $AC=18$ cm would make $EC=8$ cm and $\frac{AE}{EC}=\frac{10}{8}=1.25$, nowhere near $\frac{5}{8}=0.625$. Nothing forbids $AE$ from exceeding $AD$ (they lie on different sides), $DE$ is always shorter than $BC$ here, and repeated side lengths are perfectly legal.