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.

 

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 of parallelogram ABCD. Use this result to show that when ABCD becomes a rectangle, ∠EJD = 90o. (Note that this result also holds in the specific case when the sides of the rectangle are in the golden ratio; i.e. ABCD is a golden rectangle.)

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Michael de Villiers, created 9 July 2018, updated 22 Oct 2018.