## 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.