In quadrilateral ABCD, named in order, the angle at B and the angle at D are both right angles. What must be true of the four vertices?

The angles at B and at D are opposite angles of the quadrilateral and they add to 90° + 90° = 180°, which is exactly the test for a cyclic quadrilateral, so the four points always lie on one circle. Which chord is the diameter? The right angle sits at B and at D, and both of those angles stand on the segment AC, so AC subtends 90° at two points of the circle — an angle in a semicircle — making AC the diameter. BD is merely an ordinary chord. A rectangle would need all FOUR angles to be right angles, and no condition such as AB = AD is needed for the concyclicity.