A group of 2n people are seated around a circular table. In how many ways can they shake hands simultaneously so that every participant shakes hands with another and no handshakes cross other handshakes.

My strategy was to set up a correspondence between a set of handshakes and one way of triangulating a 2n-gon.
Then a handshake corresponds to either an edge or a diagonal, and the answer is C2(n-1).
But I'm a little unsure of my answer, and I'd like to know if I did it the right way.
I also have two questions.

#1. the meaning of "every participant shakes hands with another"
I took this to mean that no vertex (participant) is disconnected; is this the right interpretation?
At first I thought this meant each participant had to shake hands with all the other participants, but upon making a drawing, I found it impossible.

#2. connection between parentheses
Since the problem said 2n people instead of just n, I wondered if there is a way to set up a correspondence between handshakes and a string of (correctly arranged) parentheses.