A quadrilateral is called cyclic when a single circle can be drawn through all four of its vertices. Which one of these is NOT necessarily cyclic?

A quadrilateral is cyclic exactly when its opposite angles are supplementary. In a rhombus the opposite angles are EQUAL, say θ and θ for one pair, so the condition becomes θ + θ = 180°, forcing θ = 90° — that is, forcing the rhombus to be a square. So a rhombus that is not a square can never be cyclic. Rectangles and squares have four right angles, and in an isosceles trapezium each pair of opposite angles adds to 180° because the base angles at each parallel side are equal; all three of those are always cyclic.