1) To lead on the march, to marry a wife or command. In practice, the word participates in the phrase “Ducunt volentem fata, nolentem trahunt”. It says that destiny leads these who wish it, and drags these who are unwilling to follow it.
2) To draw, shape, construct / (time) spend, delay.
3) To charm, influence, mislead, draw in.
4) to calculate, count, reckon, esteem, consider (for examples, a chances in success on prospective lawsuit).
Synonyms of “duco” are “cogito”, “intueor”, and “meditor”.