The key-idea is that a symmetry is a product of two reflexions whose axes are orthogonal and intersect at the center of symmetry. Besides these axes, representing the symmetry, can be turned about their intersection point, so that one of them is parallel to any given direction (the other axis then assuming the orthogonal to that direction). This implies easily that the product of two symmetries is a translation. Thus dividing the product of 2N symmetries in pairs (f1*f2)*(f3*f4)* ... etc. we get a product of N translations (blue segments in our picture), which is a translation. The statement is that this translation (GM in our case) is , in general, non-zero (see SymmetriesOnVerticesEven.html ).

The case of polygons with odd number of sides (unique solutions to Carnot's problem) is handled in the file: SymmetriesOnVerticesOdd.html .

The case of symmetric polygons with even number of sides (infinite many solutions to Carnot's problem) is handled in the file: SymmetriesOnVerticesEven2.html .

The case of non-symmetric polygons with even number of sides and infinite many solutions to Carnot's problem is handled in the file: NonSymmetricClosed.html .

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