What is the probability that a (possibly degenerate) triangle made by three randonly chosen points on the perimeter of an n-gon contains the centre of an n-gon?
For a square, there is a $\frac{1}{16}$ chance that the points are in configuration a, $\frac{3}{16} $ for configuraion b, and $\frac{3}8$ for c and d. The probility that the points contain the center is $0$ for a and c, $\frac{1}3$ for b (since the center is contained iff one point is on each side of the line TF1 and arbitrarily taking the square to have unit sides yields $2\int_0^1 a-a^2 \mathrm{d} a=\frac{1}3$) and $\frac{1}2$ for d (center contained iff B1 is the opposite side of the line through D1H1 to C1, $\int_0^1 a \mathrm{d}a=\frac{1}2$).Therefore, if I have somehow not made an error, the probability is $\frac{1}4$.
[edited] The limiting case of a circle is $\frac{2}{\tau}\int_0^{\frac{\tau}2}\frac{a}{\tau}\mathrm{d}a=\frac{1}4$ (using $\tau=2\pi$ just to be controversial)