Nested Eggs: Where Brianchon, Pascal and Poncelet Meet

Brianchon’s and Pascal’s theorems are two famous theorems in geometry that are known to be dual to each other. This study explores if new properties can be found when they are applied to bicentric hexagons.
This study started by constructing two families of hexagons. Firstly, we extended six sides of a bicentric hexagon into the form of a hexagram followed by connecting its six vertices into a new hexagon. This procedure called operator S. Secondly, we drew tangents to the vertices of bicentric hexagon. The intersections of adjacent tangents will form a new hexagon. This procedure called operator T. By applying the polarity principle, we found that all bicentric hexagons constructed under operator S or T are Poncelet hexagons. Furthermore, these two operators are commutative.

Scientific field






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