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Journal for Geometry and Graphics 25 (2021), No. 2, 205--225
Copyright Heldermann Verlag 2021

Poncelet Porisms in Hyperbolic Pencils of Circles

Ronaldo Alves Garcia
Universidade Federal de Goiás, Goiânia, Brazil

Boris Odehnal
University of Applied Arts, Vienna, Austria

Dan Reznik
Data Science Consulting Ltd., Rio de Janeiro, Brazil

The usual Poncelet porisms deal with polygons which are inscribed into one conic and circumscribed to another conic. A more general form of Poncelet porisms considers polygons whose sides are tangent to more than one conic of a pencil of conics. We shall study the case of poristic triangles inscribed into a circle c1 with sides tangent to two further circles c2, c3 and all three circles shall be contained in a hyperbolic pencil of circles. In order to allow poristic triangle families, the radii and central distances of the circles are subject to certain algebraic relations. The main contribution of this article is to derive these relations for two special cases: In the first case, only proper circles are involved, while in the second case, we allow one circle to shrink to a point. We also pay attention to traces of triangle centers of the poristic families. Finally, we also provide closing conditions for three more types of circle pencils.

Keywords: Poncelet transverse, hyperbolic pencil of circles, closing condition, point orbit, 3-periodic billiard.

MSC: 51M04; 51D30.

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