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    Some Poncelet invariants for bicentric hexagons

    96605.pdf (249.3Kb)
    Access Status
    Open access
    Authors
    Keady, Grant
    McAndrew, A.
    Date
    2024
    Type
    Conference Paper
    
    Metadata
    Show full item record
    Citation
    Keady, G. and McAndrew, A. 2024. Some Poncelet invariants for bicentric hexagons. In Proceedings of the Asian Technology Conference in Mathematics 2024, 8th Dec 2024, Yogyakarta, Indonesia.
    Source Conference
    Asian Technology Conference in Mathematics 2024
    Additional URLs
    https://atcm.mathandtech.org/ElectronicProceedings.htm
    ISSN
    1940-4204
    Faculty
    Faculty of Science and Engineering
    School
    School of Elec Eng, Comp and Math Sci (EECMS)
    URI
    http://hdl.handle.net/20.500.11937/96841
    Collection
    • Curtin Research Publications
    Abstract

    Tangential polygons are (convex) polygons for which every side is tangent to an inscribed circle. Cyclic polygons are those for which every vertex lies on a circle, the circumcircle. Bicentric $n$-gons are those which are both tangential and cyclic. Every triangle is bicentric. Bicentric quadrilaterals are those for which the sum of the lengths of opposite sides is the semiperimeter and for which opposite angles sum to $\pi$. Here we give some results pertaining to invariants of (convex) bicentric hexagons.

    A remarkable result of Poncelet is that if one has a pair of circles admitting a bicentric $n$-gon, then for every point on the circumcircle can be a vertex for a bicentric $n$-gon. This is illustrated in the animation at\\ \verb$https://mathworld.wolfram.com/PonceletsPorism.html$

    The animation indicates that, along with the incentre and circumcentre, the point of intersection of the principal diagonals of a $2m$-gon is invariant under the motion. Such invariants -- here called Poncelet invariants -- have been studied for two centuries, in particular for triangles and bicentric quadrilaterals. We present results, for bicentric hexagons, that various combinations of distances between vertices - lengths of diagonals and of sides - are invariant.

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