(An analogous investigation of the below for a triangle gives us a generalization of the celebrated triangle of Feynman).
Investigate
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 to Parallelogram). 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.
Related Links
Feynman's Triangle: Some Generalizations & Variations
Feynman Triangle & Parallelogram Variations
Area Parallelogram Partition Theorem
Some Parallelo-hexagon Area Ratios
Area ratios of some polygons inscribed in quadrilaterals and triangles
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
Euclid 1-43 Parallelogram Area Theorem
The theorem of Hippocrates (470 – c. 410 BC)
Triangle Area Formula in terms of angles, r & R
Cross's (Vecten's) theorem & generalizations to quadrilaterals
Area Formula for Quadrilateral in terms of its Diagonals
Maximum area of quadrilateral problem
The Equi-partitioning Point of a Quadrilateral
The Orthocentre Quadrilateral of a Quadrilateral
Minimum Area of Miquel Circle Centres Triangle
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
External Links
Richard Feynman (Biography at Wikipedia)
One-seventh area triangle
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Created by Michael de Villiers, Sept 2009; modified 26 July 2021; updated 9 Dec 2023.