Lets break this circle and make it a straight line of length $2\pi$ at the point $(1,0).$ Now, $(1,0)$ is present at both ends and if we choose $3$ random points along this line, we are dividing this line into $4$ parts with all their lengths equally likely to be $\frac{2\pi}{4}.$ Since, two of the parts are favorable, our required expectation will be $\dfrac{1}{4} \times 2 \times 2\pi = \pi.$