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answering your related question:
The circumcircle of the triangle is not necessarily the minimum bounding circle of the polygon. To see this, consider a very flat isosceles triangle, say with vertices at (0,0), (10,0) and (5,1). The minimum bounding circle has center (5,0) and radius 5, but this circle doesn't touch the vertex at (5,1), so it's not the circumcircle. (The circumcircle has center (5,-12) and radius 13)
edit:
Choosing the smaller of the circumcircle or the circle containing the antipodal points of the diameter of the polygon also doesn't suffice, because it is possible to construct polygons that have points outside the circumcircle of the maximal triangle. Consider the pentagon with vertices at:
(-5, 0)
(-4, -1)
( 5, 0)
( 4, 1)
(-4, 1)
The maximal triangle has vertices at (-4,-1), (5, 0), and (-4, 1). Its circumcircle does not include the point at (-5, 0).