A Special Case of the Random Triangle Problem

jd2718's fascinating discussion regarding selection of a random triangle a couple of weeks ago led me to consider the strategy of particularization, aka, 'consider a special case' or 'make it simpler', approaches I typically suggested to students when faced with a problem that was confusing, vague or overly general. Instead of phrasing it in terms of geometric probability, I've reduced it to a much simpler problem, followed by a locus problem, a topic perhaps currently underemphasized.

First a 'simpler problem' fro your geometry students:

Consider points A(-3,0) and B(3,0) in the coordinate plane. If C(x,y) is a point in the plane such that angle ACB is a right angle, determine the value of x² + y².
Note: Many capable students would rush into the distance formula and the Pythagorean Theorem, but there is another approach that is less algebraically cumbersome!

Now for more generalization...

The following questions regard points in a single plane. If A and B are 2 arbitrary points in a plane, d units apart, determine the locus of all points C in that plane such that
(a) Angle ACB is right
(b) Angle ACB is acute
(c) Angle ACB is obtuse

Notes:
(i) Why do you think I chose to focus on angle ACB rather than triangle ACB as in Jonathan's discussion?
(ii) If students are unfamiliar with the term locus, rephrase as "Describe the set of all points C such that..."
(iii) Will this question lead directly to solving the general probability question raised by zac or Jonathan? Probably not, but I was thinking more in terms of accessibility for most geometry students. The open-endedness of the original triangle problem is highly instructive and can lead to profound considerations but that can be tackled later...