Draw the perpendicular bisectors of any two sides of the cyclic quadrilateral. Their intersection gives the circumcentre. Then observe whether this point lies inside or outside the quadrilateral.
Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out?
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The best way is to use the perpendicular bisectors of the sides. Draw the perpendicular bisectors of any two sides of the cyclic quadrilateral. Their point of intersection is the centre of the circumcircle because the centre is equidistant from the endpoints of each chord. After locating this point, check its position with respect to the quadrilateral. If it lies within the quadrilateral, the centre is inside; otherwise, it is outside.
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