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).
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.
Michael de Villiers, Sept 2009.