It is well-known, as shown below, that a regular pentagon ABCDE has the following ratios equal to the famous golden ratio, φ = (√5 + 1)/2 = 1.61803...
The Parallel-pentagon and the Golden Ratio
Far less well-known is that the above ratios also hold for a parallel-pentagon - defined as a convex pentagon with each diagonal parallel to the side with which it has no vertex in common (Gilbert, Krusemeyer & Larson, 1993). (For example, ABCDE is a parallel-pentagon if diagonal AC is parallel to side DE and similarly for the other four diagonals).
Below is a dynamic parallel-pentagon ABCDE with some ratios of line-segments measured & displayed. (Sides and diagonals that are parallel are given in the same colour).
1) What do you notice about the ratios?
2) Drag any of the red vertices & repeat 1) a few times.
3) Can you drag the figure into the shape of a regular pentagon?
Parallel-pentagon and Golden Ratio
Challenge
4) Can you explain why (prove that) your observations above about the ratios are true?
5) Can you construct a 'dynamic' parallel-pentagon using dynamic geometry software?
(In other words, one that you can drag like the one above, but the diagonals remain parallel to the corresponding sides as defined above).
Hints: Read Gilbert et al and/or Bogomolny in the references below.
References
Gilbert, G.T.; Krusemeyer, M.I. & Larson, L.C. (1993). The Wohascum County Problem Book. The Mathematical Association of America, Dolciani Mathematical Expositions No. 14, pp. 80-82.
Bogomolny, A. (Undated). Golden Ratio in an Irregular Pentagon, Construction II. Cut the Knot Website.
Related Links
Golden Ratio Parallel-hexagon
Golden Quadrilaterals (Generalizing the concept of a golden rectangle)
Cyclic Kepler Quadrilateral Conjectures
A Fibonacci Generalization - Kendal's theorem
Number Patterns: What number comes next?
Some External Links
Golden Ratio in Regular Pentagon
Golden Ratio in an Irregular Pentagon, Construction II
Golden Ratio in an Irregular Pentagon
'Golden Rectangle'
Golden Ratio (Wikipedia)
Golden Ratio (Cut the Knot)
Myth-busting the Golden Ratio
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Created by Michael de Villiers with WebSketchpad, 25 Dec 2023.