2019 AMC 10A Problem 6

Note that if a point is equidistant from all the vertices, then that point is the center of the shape’s circumcircle. The question then becomes which of these shapes is cyclic (has a circumcircle). One condition that we can use is that opposite angles are supplementary. Clearly, a square and rectangle that is not a square work (opposite angles are right, adding up to $180^{\circ}$). A rhombus that is not a square does not work, since opposite angles are equal, but they are not $90^{\circ}.$ A parallelogram that is not a rectangle or a rhombus faces the same problem as above, making it not cyclic as well. An isosceles trapezoid that is not a parallelogram by definition has supplementary opposite angles, making it cyclic. Thus, C is the correct answer.