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• Second, discuss the strategy of inspecting inscribed angles. Here is an image that highlights inscribed angles and their associated central angles:
Ask students,
• “How does the proof involving isosceles triangles change if the circumcenter is outside the quadrilateral?” (If the circumcenter is outside the quadrilateral, then the base angles of the isosceles triangles are no longer the same as the angles of the quadrilateral. The argument doesn’t work quite the same way.)
• “How does the proof involving inscribed angles change if the circumcenter is outside the quadrilateral?” (If the circumcenter is outside the quadrilateral, then the arc traced out by one of the angles will take up almost the whole circle, and the opposite angle will trace out a very small arc. The argument is still valid with no modi cations, though. The two inscribed angles together trace out the entire circumference, regardless of the location of the circumcenter.)
It is not necessary to ask students to prove the theorem with isosceles triangles in the cases where the circumcenter is not inside the cyclic quadrilateral, although it is o ered as an extension.
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Teacher Guide