Feynman Parallelogram Generalization

For a parallelogram ABCD in the plane, if each vertex is joined to the point (1/p), (p >= 2) along the alternate side (measured say anti-clockwise), then what is the relationship between the area ABCD and the area EFGH (the inner parallelogram formed by these lines)?

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Feynman parallelogram generalization: 1/2 division

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Feynman parallelogram generalization: 1/3 division

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Feynman parallelogram generalization: 1/4 division

1. Read my article at Feynman's triangle: Some feedback and more

2. Can you generalize further for sides divided into different ratios?

Note: In the (1999/2003) Prefaces of my Rethinking Proof with Sketchpad book by Key Curriculum Press, it is decribed how the first result for the midpoints was investigated for any convex quadrilateral by a class of mine in 1995, leading to the eventual conjecture by a student, Sylvie Penchaliah, that 1/5 area ABCD >= area EFGH > 1/6 area ABCD, and equality holds when EFGH is a trapezium as shown in Sylvie's Theorem. A proof of the result by Avinash Sathaye, Carl Eberhart and Don Coleman from the Univ. of Kentucky in 2002 is available at Coleman proof. Another proof and further extension by Marshall, Michael & Peter Ash in 2008 in an article submitted to the Mathematical Gazette can be found at Ash proof. Sylvie's Theorem also appears as a conjecture in a paper by Keyton, M. (1997). Students discovering geometry using dynamic geometry software. In J. King & D. Schattschneider (Eds.), Geometry turned on! Dynamic software in learning, teaching and research (pp.63-68). Washington, DC: The Mathematical Association of America. Downloadable Sketchpad files from the Keyton paper are available at Keyton GSP files. Keyton's sketch no. 6 corresponds to the general construction.


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Michael de Villiers, Sept 2009.