(An analogous investigation of the below for a triangle gives us a generalization of the celebrated triangle of Feynman).

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)?

Feynman parallelogram generalization

**Challenge**

Can you prove the above result? Can you prove it in more than one way?

**Further Generalization**

Can you generalize further if the sides of the parallelogram are divided into different ratios?

**References**

De Villiers, M. (2005). Feedback: Feynman's Triangle (extended). *The Mathematical Gazette*, 89 (514), March, p. 107.

Hindin, H.J. (2010). From Feynman to Fibonacci And More. Paper presented at the Fifth Annual Spuyten Duyvil Mathematics Conference, St. Francis College, Brooklyn, New York, April 24, 2010.

**Notes**

1) In the (1999/2003) Preface 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*.

2) 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.

3) A computer assisted proof of Sylvie's Theorem by Avinash Sathaye, Carl Eberhart and Don Coleman from the Univ. of Kentucky in 2002 is available at *Coleman proof*.

4) Another proof and further extension of Sylvie's theorem by Marshall, Michael & Peter Ash in 2007 in an article in the *Mathematical Gazette*, 93(528), can be found at *Ash proof*.

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Created by Michael de Villiers, Sept 2009; modified 26 July 2021.