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[–] hans_wormhat [S] ago  (edited ago)

To even put forth such an argument is to beg the question [perhaps suggests] that the circle is a polygon to begin-with.

But to discriminate the circle from the set "Reuleaux polygons" because it is equal to the circle which circumscribed the even-sided polygon whose widest points give us that curve of constant width which is our circle, is to disprove the assumption begging the question [suggested] within the presented argument.

For to say that the curve of constant width which finds all its points upon the circle that constructed it is not a Reuleux polygon is to also say that a construction of points equidistant from a centre which finds all these points along a given circle is to measure the thing against itself.