Euclidean and Non-Euclidean Geometry – Fall 2007 Dr. Hamblin

Presentation: Cyclic Quadrilaterals In this presentation, you will investigate a special class of quadrilaterals known as “cyclic quadrilaterals.” By definition, a cyclic quadrilateral is one where all four vertices lie on a single circle.

Using Sketchpad, investigate cyclic quadrilaterals and verify the following facts (you will prove these later):

The opposite angles of a cyclic quadrilateral are supplementary.

The perpendicular bisectors of the sides of a cyclic quadrilateral are always concurrent at the center of the circle.

Proofs

In order to prove these statements, we will need to use the Inscribed Angle Theorem (p. 270 in your book). This theorem states that the measure of an inscribed angle is equal to half of the measure of its intercepted arc.

As part of your presentation, you should include the following.

Explain briefly why the Inscribed Angle Theorem is true.

Use the Inscribed Angle Theorem to prove that the opposite angles of a cyclic quadrilateral are supplementary.

Explain why the perpendicular bisectors of each of the four sides passes through the center of the circle. Draw a line segment connecting opposite vertices to split the quadrilateral into two triangles. Then use the result of Problem #20 from Section 3.3.

Converse

So far, you have proved that if a quadrilateral is cyclic, then its opposite angles are supplementary. Now we will prove that if a quadrilateral has the property that a pair of opposite angles is supplementary, then it must be cyclic.

We will prove this by contradiction: Suppose that quadrilateral has the property that opposite angles are supplementary, specifically that and are supplementary. However, suppose also that is not cyclic. Follow this outline to complete the proof.

Since is not cyclic, the circumcircle of does not pass through . For now, assume that lies outside of this circle. We will consider the other case later. Euclidean and Non-Euclidean Geometry – Fall 2007 Dr. Hamblin

Since lies outside the circle, either or meets the circle. Without loss of generality, suppose meets the circle at . (You should use Sketchpad to illustrate the case where does not meet the circle.)

What kind of quadrilateral is ?

From what you already proved about cyclic quadrilaterals, what is the relationship between and ?

What do you already know about the relationship between and ? What does this tell you about and ?

Draw segment , and use the Exterior Angle Theorem to reach a contradiction.

You should now consider the case where is inside the circle. The argument should be similar, involving the facts you have already proved about cyclic quadrilaterals and the Exterior Angle Theorem.

Further Investigation: Cyclic Polygons

We can extend the definition of cyclic to pentagons, hexagons, and so on. What can you say about the measures of the angles of these kinds of polygons? Make a conjecture and prove that it is correct.