Defining the diameter of a convex polygon as the maximum possible distance between all pairs of vertices, can we conclude that the convex polygon is inscribable (i.e has all its sides as chords of a circle) if the diameter isn't the diameter of the minimum bounding circle, in which case the circumscribed circle is the minimum bounding circle?
2026-03-28 16:19:31.1774714771
When is a convex polygon inscribable?
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It seems you want "the diameter is the diameter of the minimum bounding circle". Take a long skinny rectangle, which is inscribable. Now make a pentagon by bending one long side out just a little bit. No longer inscribable-the circumscribed circle has not changed.
Added: a figure is below. The long rectangle and the pentagon have the same diameter, but the pentagon is not inscribable