Draw now the conic (c') tangent to the first five sides of the hexagon: {AB, BC, CD, DE, EF}. Conic (c') is also tangent to the sixth side FA exactly when the lines connecting opposite vertices meet at a point. This is again a consequence of Brianchon's theorem.

Now, if the opposite sides are parallel and lines joining opposite vertices pass through a point O, then O is center of symmetry of the hexagon. To see this apply Desarque's theorem on two

Disregarding the intervening conics we arive at the result, that a hexagon with parallel sides is symmetric exactly when the lines joining opposite vertices pass through a point, which is then the center of symmetry.

The file HexagonsSymmetricRepeat.html displays a figure and some questions concerning an infinite sequence of hexagons, one inscribed into the other, initiated by an arbitrary symmetric hexagon.

Desargues.html

HexagonsSymmetricRepeat.html

Pascal.html

Symmetric_hexagons.html

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