Similar Parallelograms: A Generalization of a Golden Rectangle property

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

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 = 90o.
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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Michael de Villiers, created 9 July 2018 using WebSketchpad, updated 22 Oct 2018; 20 Feb 2023.