Here is what I've come up with:
Instead of viewing the circle as a curve, I decided to think of it as a regular n-gon, with n approaching infinity. That way, it will be easier to count possibilities. And once we've come up with the expression for probability in terms of n, we can obtain the limit when n approaches infinity.
We position the points A, B and C on the vertices of the n-gon.
So here goes... For each of the points A, B and C, there are n ways of positioning on the n-gon. That is a total of n^3 ways of positioning the three points on the n-gon.
Now tackling the "ABC is acute-angled" condition...
A can be positioned n ways on the n-gon.
C can be positioned (n-2) ways on the n-gon. That is, C can be positioned on any vertex of the n-gon except on two positions: the current position of A and the vertex directly across A.
The vertex directly across A is eliminated as a possible position for C because AC will be the diameter, the intercepted arc AC is a half-circle, which means ABC is automatically a right angle.
Now I'm stuck with how to position B. Obviously, B must be located on the major arc AC, in order for ABC to intercept the minor arc.
Now the number of possible positions for B is determined by the positions of A and C relative to each other.
The closer A and C are to each other, the larger the major arc and the more possible positions B can have.
Any ideas on how to write an expression for the number of positions of B?