convex quadrilateral
in [Math]
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From: Han de Bruijn on 5 May 2010 06:57 On 4 mei, 23:02, Ken Pledger <ken.pled...(a)mcs.vuw.ac.nz> wrote: > In article > <39b1b326-77bc-45d6-842b-5907ff756...(a)d19g2000yqf.googlegroups.com>, > > quadratic <quadra...(a)juno.com> wrote: > > How does one prove that in any convex quadrilateral, the quadrilateral > > formed by the midpoints of the sides is a parallelogram? > > You can probably track down several ways of proving it by > searching for "Varignon's Theorem". Incidentally, the quadrilateral > needn't be convex, and it needn't even be in one plane. > > Ken Pledger. Does somebody realize how interesting this elementary geometrical fact is for applications in Numerical Analysis? Without it, there would be no simple (least squares) finite element for 2D ideal flow. It's in: http://hdebruijn.soo.dto.tudelft.nl/jaar2010/ [ namely: ] http://hdebruijn.soo.dto.tudelft.nl/jaar2004/vierhoek.pdf http://hdebruijn.soo.dto.tudelft.nl/jaar2004/nlrlsfem.pdf Search for "Numerical Method for 3D Ideal Flow", and stuff surrounding that item. Does somebody know what the generalization of that parallelogram is in three dimensions? Well, the generalization of a general quadrilateral is a hexahedron .. Hint: solution on the same "jaar2010" webpage: the inner paralellogram does _not_ generalize to a hexahedron, but it's .. Han de Bruijn |