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Circle the wagons!
wagon fort
The point from which the circle is drawn IS the center.
If you mean can a circle always be drawn round a quadrilateral so that the quadrilateral is enclosed within the circle then yes as long as the diameter of the circle is large enough. If you mean can a circle always be drawn around a quadrilateral so that it passes through all vertices then only if the opposite angles of the quadrilateral add up to 180o - such quadrilaterals are called cyclic quadrilaterals.
If you draw a line from the center of a circle to the edge of a circle, you have drawn the radius of the circle. If you draw a line from the edge of a circle through the center of the circle and on to the edge on the opposite side, you have drawn the diameter of a circle. The radius of a circle is one half the diameter of a circle.