Second, I believe I may have found two mistakes in your page entitled Inscribed and Central Angles. ( https://www.cut-the-knot.org/Curriculum/Geometry/InscribedAngle.shtml ) See if you agree. It states:

"Let A' be the point on the circle opposite point A. The proof is simplest when one of the points - B or C - coincides with A'. In this case, the inscribed angle BAC is one of the two base angles in an isosceles triangle AOA'."

But if A' is opposite A, then AOA' will always be a diameter of the circle, not a triangle. I think the Isosceles triangle we want to consider here is either AOB or AOC, the former if you choose C to coincide with A' and the latter if you choose B.

Then, the proof states:
"Two additional cases have to be considered: A' may lie on either the arc subtending the inscribed angle or the one that also contains the point A. In the former case the angle BAC is acute, in the latter it is obtuse."

But it is not A' being on one of those two arc which determines if the angle BAC is acute or obtuse, but rather the position of A itself. In this and the following discussion I think you have mixed A with A' only in talking about which of them determines the distinction between cases -- your other notation seems correct.

Am I correct here or have I gotten myself completely turned around by playing with Java applets for too long?

Thank you for the kind words and pointing out the mishap. The mistakes is main. I fixed the first right away. The second is a complete nonsense. It does not matter whether angles are acute or obtuse, but only wether A' is between B and C. This part needs a complete rewrite.

>Am I correct here or have I gotten myself completely turned
>around by playing with Java applets for too long?

You are correct. It appears that writing too many applets may affect a fellow's judgement.