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ABCD is a convex quadrilateral
A convex quadrilateral has a circumcircle only when its opposite angles add up to $$180^\circ$$, and an incircle only when the sums of its opposite sides are equal. Some convex quadrilaterals satisfy these (a square has both) and some do not (a non-square rectangle has no incircle, a non-square rhombus has no circumcircle). So none of the "always" or "never" statements holds.
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