Cyclic Quadrilaterals

Topic:
Geometry

Exploring properties of cyclic quadrilaterals

1. Drag the vertices of the quadrilateral. What kind of quadrilaterals can you form? What kind of quadrilaterals is not possible to make? Explain.

2. Drag the vertices of the quadrilateral. What do you notice about their interior angles?

Drag any of the vertices of the quadrilaterals outside or inside the circle. The new quadrilateral is no longer a cyclic quadrilateral. Does the conjecture hold for non-cyclic quadrilaterals? Drag different vertices again to check your conjecture.

Check all that apply

Drag the vertices of the quadrilateral to look for the relationship between the quadrilateral and the bisectors of the angles. Do you notice anything?

Drag the vertices of the quadrilateral to look for the relationship between the quadrilateral and the perpendicular bisectors. Do you notice anything? Make a conjecture about the relationship you observe:

Explaining the conjectures

In the preceding section, you formulated the following conjecture: Opposite angles of a cyclic quadrilateral are supplementary or their sum is equal to 180 degrees (as long as the quadrilateral is not “crossed”). You also found out that this is true because the quadrilateral is inscribed in the circle.

Focus your attention on one of the pair of opposite angles. What kind of angle is this?

Drag points A, B, C. What is the relationship between an inscribed angle and the arc angle that you observe?

Use this result to build a logical progression of statements to explain why in a cyclic quadrilateral the opposite angles are supplementary (Hint: In your original sketch, construct the chord AC).

In the preceding section, you formulated the following conjecture: The perpendicular bisectors of the sides of a cyclic quadrilateral always remain concurrent at the center of the circle. This center is called the circumcenter of the cyclic quadrilateral. Build a logical progression of statements to explain why in a cyclic quadrilateral the perpendicular bisectors remain concurrent at the center of the circle (Hint: First argue why each perpendicular bisector goes through the center. Construct radii to help.)