What is probability that $3$ random point in $2n$ reguler polygon contains center of gravity
I contain the diagram to make up my explanation. Left one is contain the center of gravity,while right one doesn't.
$$\binom{2n}{n}$$ is the whole case and I found that when $n=1$,Probablity $P_{1} =$ Not defined.And $n=2$,$P_{2}=1$. I also get $P_{3}=\frac{7}{10}$ and $P_{4}=\frac{4}{7}$, but I can't certain about this.
Also I found $$\lim_{n\rightarrow\infty}P_{n} =\frac{1}{4}$$
Plz help me to find $P_n$