In the previous post, we analyzed the relationships among various quadrilaterals, organizing our findings in a flowchart. Now we will examine the effect of throwing a curveball into the mix- a cyclic quadrilateral. By definition, a cyclic quadrilateral is a four-sided polygon whose vertices all lie on a given circle (an example is pictured above). By constructing a generic cyclic quadrilateral using GeoGebra, a student can manipulate the placement of the vertices along the circle in order to uncover the specific types of quadrilaterals that can be formed from a cyclic quadrilateral. Upon investigation, a student will deduce that the following quadrilaterals are *inherently* cyclic:

- Isosceles Trapezoid
- Rectangle
- Square
- Right Kite

In fact, upon deeper examination, the only way in which a trapezoid, parallelogram, rhombus, or kite can be a cyclic quadrilateral is in its respective special case listed above. For example, a kite will only be cyclic if it is the special case of a right kite (kite whose congruent angles measure 90 degrees). These findings are summed up in the following new flowchart:

Thus, the cyclic quadrilateral points to the isosceles trapezoid and right kite (and by transitivity, the rectangle and square) because each of these polygons is always a cyclic quadrilateral. By contrast, the cyclic quadrilateral does not point to the trapezoid, parallelogram, rhombus, or kite because these polygons are only cyclic in the special cases already covered.

Extension assignments such as this one help students build critical thinking skills as it forces them to evaluate everything they have learned about quadrilaterals thus far. By extending the flowchart a student has created, he or she will have to reconsider the relationships among quadrilaterals, thereby strengthening his or her relational understanding of the topic.

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I agree with what you said regarding which quadrilaterals can be cyclic. I also stated that a parallelogram and rhombus can only be formed as a rectangle and square. Thus, not all parallelograms and rhombuses can be constructed from a cyclic quadrilateral.

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I agree that some quadrilaterals can be cyclic, but my flowchart is different from yours. I connected cyclic to kite and trapezoid because a cyclic quadrilateral can make both a kite and a trapezoid. I looked at the investigation differently from you. Thus, I am not disagreeing with you because I see your reasoning behind your flowchart.

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I like the way your flowchart is set up. However, I feel like cyclic quadrilaterals should be connected to kites and trapezoids.

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