(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
Areas Activity from Rethinking Proof
Feynman's Triangle: Some Generalizations & Variations
Feynman Triangle & Parallelogram Variations
Sylvie's Trapezium Area Inequality Theorem
Area Parallelogram Partition Theorem
Some Parallelo-hexagon Area Ratios
Vreken's sum of triangular areas in a regular hexagon
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)
Sums of alternate triangular areas in regular & semi-regular side-2n-gons (Application of Viviani)
Euclid 1-43 Parallelogram Area Theorem
The Area of an Orthodiagonal quadrilateral
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
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
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
Sums of alternate triangular areas in regular & semi-regular side-2n-gons (Application of Viviani)
External Links
Richard Feynman (Biography at Wikipedia)
One-seventh area triangle
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Created by Michael de Villiers, Sept 2009; modified 26 July 2021; updated 9 Dec 2023; 18 March 2026; 24 August 2026.