Catalan number question (possibly)

**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 C_{2(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.