Anywhere in the yellow region (and/or the gray arcs) below one can
choose the third point C for a triangle subject to the condition
that neither sides AC nor BC exceeds side AB.

Triangle Possibilities

Once chosen, the resulting triangle can be classified by its angles
(obtuse, acute, right) or its sides (scalene, isosceles, equilateral).
The figure below shows these possibilities. Since the figure is
top-bottom symmetric, the same categories exist for the bottom half
of the figure as the top half.

For example, if point C is chosen on the blue circle (except at points
A and B), then the resulting triangle ABC is a right triangle and a
scalene triangle unless point Q is chosen, in which case triangle ABC
is an isosceles right triangle.

Triangle Categories
If C is selected on the gray arcs or lines outside the blue circle,
then the triangle (isosceles) is acute also.
If C is selected on the gray line inside the blue circle, then the
triangle (isosceles) is obtuse also.

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Thomas M. Green