The curve obtained by adjoining a diameter to a semicircle has no inscribed regular pentagon.

To see this, first note that no more than two of the vertices of a regular pentagon could be placed on the diameter segment. So, if this curve were to have an inscribed regular pentagon, at least three corners of the pentagon would have to lie on the semicircle. But any three vertices of a regular pentagon determine the circle that contains all five vertices, and no three of them lie in any half of that circle. So, it is impossible to place a regular pentagon with three or more of its vertices on the semicircle.