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?

To determine if the centre of the circumcircle of a cyclic quadrilateral lies inside or outside the quadrilateral without drawing it, we need to examine the measures of its interior angles.

Condition for Circumcentre Location:

Circumcentre Inside: The centre of the circumcircle lies inside the quadrilateral if and only if all four interior angles of the quadrilateral are acute (less than 90°).

Circumcentre Outside: The centre of the circumcircle lies outside the quadrilateral if and only if at least one interior angle of the quadrilateral is obtuse (greater than 90°).

Circumcentre on an Edge: If one of the interior angles is a right angle (90°), the centre of the circumcircle lies at the midpoint of the diagonal opposite that angle.

The best way to find out is to:

Measure or calculate the four interior angles of the cyclic quadrilateral. Let these angles be ∠A, ∠B, ∠C, ∠D.

Check the type of each angle:

If ∠A < 90°, ∠B < 90°, ∠C < 90°, and ∠D < 90°, then the circumcentre is inside the quadrilateral.

If any of ∠A, ∠B, ∠C, ∠D is greater than 90°, then the circumcentre is outside.

If one angle is exactly 90° and the others are acute, the circumcentre is on the midpoint of the opposite side.