The question: a scalene triangle determines exactly how many circles, where only the vertices of the triangle are uniquely used, or the body of the triangle in general (i.e. no arbitrarily specified points, such as the midpoint of any of the sides, are used to determine a circle)?

The answer is surprisingly high! Key thoughts: a circle can be uniquely determined by 3 non-collinear points (3 points), or by a diameter (requiring 2 points), or by a center and a radius (also 2 points)[seems a little like the ID's for getting a driver's license, doesn't it?]. Sounds simple -- until you realize that geometrical constraints can be used to yield additional points to use beyond just the three vertices [come to think of it, can it just be a coincidence that your driver's test included a *3-point* turn?].

So, there are inscribed and circumscribed circles (total, 2);

circles centered at each vertex and using each side as a radius (total, 6);

circles using each side as a diameter (note, the midpoint is incidental in this!) (total, 3);

circles using each side as a chord, such that the circle is tangent to an adjacent side at the more acute corner (i.e. an arc of the circle is inscribed within the triangle) (total, 3);

and finally, circles centered at each vertex, and tangent to the opposite side (requires both terminal angles on the opposite side to be acute, otherwise is included already on the list so far): (either 1 or 3, depending on the triangle!)

Grand total = 15 or 17, depending on the triangle.

Color the pieces in nicely, and you'll have some memorable artwork, too! (do it on Paint, with bright contrasting colors, and your eye will do color-perception somersaults scanning over it!)

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