## Sylvie's Theorem

For a convex quadrilateral *ABCD* in the plane, if each vertex is joined to the midpoint along the alternate side (measured say anti-clockwise), then 1/5 area *ABCD* >= area *EFGH* > 1/6 area *ABCD*, and equality holds when *EFGH* is a trapezium (*EFGH* is the inner quadrilateral formed by the constructed lines).

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Sylvie's Theorem

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The equality in Sylvie's Theorem

**Note**: In the (1999/2003) Prefaces of my *Rethinking Proof with Sketchpad* book by Key Curriculum Press, it is described how in 1995, a student of mine, Sylvie Penchaliah, made the conjecture that 1/5 area *ABCD* >= area *EFGH* during a class investigation. 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.