"They present their idea."
Translation:De presenterar sin idé.
It's either Hon tvättar händerna or Hon tvättar sig om händerna, but you're right that we often use the definite when English would use a possessive. (I always link to this discussion so I'm adding a link there here too: https://www.duolingo.com/comment/6014446 for any users who haven't seen it already)
I think it's not that obvious that an idea is your own – the perfect example is body parts, where you really are most likely to speak about your own ones, like Jag borstar tänderna = I brush my teeth, it's less likely that you would brush anybody else's teeth. If somebody said De presenterar idén, I would not assume that it was their own idea.