Given a parallelogram *CFED* drawn similar to a parallelogram *ABCD* as shown, explain why (prove that) *GHIJ* is cyclic. Select and drag any of the three red vertices to dynamically change the shape of the two similar parallelograms.

Similar Parallelograms form a Cyclic Quadrilateral

**Explore**

Explore the figure by dragging any of the red vertices. Also drag the parallelogram until it becomes a rectangle.

**Exercise**

1) Can you explain **why** (prove that) *GHIJ* is cyclic? Can you do it in more than one way?

2) Determine the value of ∠*EJD* in terms of the angles formed by the diagonals at the vertices of parallelogram *ABCD*. Use this result to show that when *ABCD* becomes a rectangle, ∠*EJD* = 90^{o}.

3) Click on the 'Link to Similar Rectangles' button to navigate to the specific case when *ABCD* is a rectangle. What do you notice about the ratios of the sides of similar rectangles *ABCD* and *CFED* to that of the ratio of the sides of the 'right kite' *GHIJ*?

4) Can you explain **why** (prove that) the ratio of the sides of the right kite *GHIJ* in 3) above are equal to those of similar parallelograms *ABCD* and *CFED*? (Note when the sides of the rectangle *ABCD* are in the golden ratio, it is a golden rectangle - drag the vertices of *ABCD* until it becomes one. It seems appropriate then to correspondingly call *GHIJ* a 'golden right kite'.)

**Note**: The right kite in 3) is also a harmonic quadrilateral since it is a cyclic quadrilateral with the products of its opposite sides equal. Read more about these quadrilaterals, their properties & applications to problems in this paper, About the Harmonic Quadrilateral, presented at a conference in Bulgaria in 2012 by Truong, Khanh & Quang from Vietnam.

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Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

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Michael de Villiers, created 9 July 2018 using *WebSketchpad*, updated 22 Oct 2018; 20 Feb 2023.