相关论文: Parabola-Inscribed Poncelet Polygons Derived from …
Poncelet polygons inscribed in a circle and circumscribed about conics from a confocal family naturally arise in the analysis of the numerical range and Blaschke products. We examine the behaviour of such polygons when the inscribed conic…
We explore properties and loci of a Poncelet family of polygons -- called here Steiner-Soddy -- whose vertices are centers of circles in the Steiner porism, including conserved quantities, loci, and its relationship to other Poncelet…
We analyze loci of triangle centers over variants of two-well known triangle porisms: the bicentric and confocal families. Specifically, we evoke the general version of Poncelet's closure theorem whereby individual sides can be made tangent…
We show that the focus of the Kiepert in-parabola remains stationary over a family of circle-inscribed Poncelet triangles which contain an equilateral triangle.
The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that…
We study families of triangles that are inscribed in a fixed circle and circumscribed about a central conic, extending the classical Chapple--Euler relation within the framework of Poncelet geometry. We establish several geometric…
We describe some three-dozen curious phenomena manifested by parabolas inscribed or circumscribed about certain Poncelet triangle families. Despite their pirouetting motion, parabolas' focus, vertex, directrix, etc., will often sweep or…
We present a theory which predicts if the locus of a triangle center over certain Poncelet triangle families is a conic or not. We consider families interscribed in (i) the confocal pair and (ii) an outer ellipse and an inner concentric…
We study pairs of conics $(\mathcal{D},\mathcal{P})$, called \textit{$n$-Poncelet pairs}, such that an $n$-gon, called an \textit{$n$-Poncelet polygon}, can be inscribed into $\mathcal{D}$ and circumscribed about $\mathcal{P}$. Here…
We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of…
The locus of the centers of mass of the family of Poncelet polygons, inscribed into a conic $\Gamma$ and circumscribed about a conic $\gamma$, is a conic homothetic to $\Gamma$.
We tour several Euclidean properties of Poncelet triangles inscribed in an ellipse and circumscribing the incircle, including loci of triangle centers and envelopes of key objects. We also show that a number of degenerate behaviors are…
We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove…
We describe four special families of ellipse-inscribed Poncelet triangles about the incircle which maintain certain triangle centers stationary and which also display interesting conservations.
We prove that over a generic Poncelet triangle family, the locus of the circumcenter of an inversive triangle is a conic. Additionally, we prove an earlier conjecture: over generic Poncelet triangles, two unique points exist which maintain…
We compare loci types and invariants across Poncelet families interscribed in three distinct concentric Ellipse pairs: (i) ellipse-incircle, (ii) circumcircle-inellipse, and (iii) homothetic. Their metric properties are mostly identical to…
In this paper, we consider conic-line arrangements that arise from Poncelet's closure theorem. We study unramified double covers of the union of two conics, that are induced by a $2m$-sided Poncelet transverse. As an application, we show…
We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a "homogeneous lamina") for degenerate Poncelet polygons. We also provide a proof for one of Dan…
We present a new proof of the necessary and sufficient condition for the existence of a triangle that is simultaneously inscribed in a circle and circumscribed about a central conic (an ellipse or a hyperbola). In the limiting case where…
We study the loci of the Brocard points over two selected families of triangles: (i) 2 vertices fixed on a circumference and a third one which sweeps it, (ii) Poncelet 3-periodics in the homothetic ellipse pair. Loci obtained include…