Symmetry, Group Actions, and Euclidean Geometry
September 26, 2005
Bailey Hall 201
Pastries and drinks will be served
Did you know that if you connect the midpoints of an arbitrary quadrilateral, you obtain a parallelogram? This simple theorem from Euclidean geometry is not hard to prove from the Euclidean axioms, and even easier if you use analytic geometry (Cartesian coordinates). However, neither the synthetic nor the analytic proof gives one any insight on what happens if the quadrilateral is replaced by a polygon with n-sides, nor what happens if you try to "invert" the theorem. In this talk, we outline an alternate approach using a group of transformations that completely explains these results, and has other applications to Geometry besides the above results on quadrilaterals. The key is to take advantage of the symmetry of the situation.
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