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 find out whether the centre of the circumcircle lies inside or outside a cyclic quadrilateral, divide the quadrilateral into two triangles by drawing a diagonal. Then, check the nature of each triangle formed.
If both triangles are obtuse-angled, the circumcentre lies outside the quadrilateral. If at least one triangle is acute-angled, the circumcentre lies inside the quadrilateral. If at least one triangle is right-angled, the circumcentre lies on the diagonal (the boundary of the quadrilateral).
Explanation
The textbook context provides the key property that for an acute-angled triangle, the circumcentre lies inside the triangle; for an obtuse-angled triangle, the circumcentre lies outside the triangle; and for a right-angled triangle, the circumcentre lies at the midpoint of the hypotenuse.
By applying this property to the two triangles formed by a diagonal of a cyclic quadrilateral, we can determine the position of the circumcentre relative to the quadrilateral. Since the quadrilateral is the union of the two triangles sharing the diagonal, the circumcentre lies inside the quadrilateral if it lies inside at least one of the triangles (i.e., at least one triangle is acute-angled). It lies outside only if it lies outside both triangles (i.e., both are obtuse-angled).
Solution Steps
Step 1: Divide the cyclic quadrilateral into two triangles by drawing any one diagonal.
Step 2: Examine each triangle to determine if it is acute-angled, obtuse-angled, or right-angled.
Step 3: Apply the circumcentre property: for an acute-angled triangle, the circumcentre lies inside; for an obtuse-angled triangle, it lies outside; for a right-angled triangle, it lies at the midpoint of the hypotenuse (the diagonal).
Step 4: If both triangles are obtuse-angled, the circumcentre lies outside both triangles, hence outside the quadrilateral.
Step 5: If at least one triangle is acute-angled, the circumcentre lies inside that triangle, hence inside the quadrilateral. If at least one triangle is right-angled, the circumcentre lies on the diagonal (boundary).