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Student Response
If there were a point that was the same distance from all four town centers, they would form a cyclic quadrilateral. That would mean the opposite angles were supplementary. Because the angles aren’t supplementary, this must be impossible.
Student Lesson Summary
We saw that any triangle can be circumscribed by a circle, but it is not always possible for a quadrilateral. If a circle can be drawn to pass through all 4 vertices of a quadrilateral, it is called a cyclic quadrilateral. We can prove that the opposite angles of a cyclic quadrilateral must be supplementary. Here is a cyclic quadrilateral with opposite angles and :
The measure of angle is labeled as radians and the measure of the opposite angle, angle , is labeled as radians. Notice that these angles are inscribed angles. This diagram shows the related central angle measures labeled as and :
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Teacher Guide