Liliana Gabriela Gheorghe
We describe all triangles that shares the same circumcircle and Euler circle. Although this two circles do not form a poristic pair of circles, we find a poristic circle "in-between" that enable to solve this problem using Poncelet porism.
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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 families.
Previously we showed the family of 3-periodics in the elliptic billiard (confocal pair) is the image under a variable similarity transform of poristic triangles (those with non-concentric, fixed incircle and circumcircle). Both families conserve the ratio of inradius to circumradius and therefore also the sum of cosines. This is consisten with the fact that a similarity preserves angles. Here we study two new Poncelet 3-periodic families also tied to each other via a variable similarity: (i) a first one interscribed in a pair of concentric, homothetic ellipses, and (ii) a second non-concentric…
When Grünbaum meets Poncelet -- Infinite Classes of Movable $n_4$ Configurations
本文将Poncelet Porism的运动特性推广到$(n_4)$构型，证明所有平凡天体构型均可动，还给出了关联离散可积系统的替代研究路径。
A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be constructed starting with any initial circle, tangent to the parent circles. What do the circles in these 1-parameter family of Steiner chains of length k have in common? We prove that the first k-1 moments of their curvatures remain constant within a 1-parameter family. For k=3,…
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 to separate in-pencil caustics. We show that despite the more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.
We give a sharp construction for twins in arbelos, based on polar reciprocity. In the process, new circles displaying arquimedean afinities came into scene.
An Affine Approach to Metric Invariants of Poncelet Polygons
本文推广Dan Reznik的几何观察结果，基于Marden定理的三角形对称参数化给出更强形式的证明，还推广了其奇数边形相关结论，并提出圆内接n边形外切于中心二次曲线时的正方形总面积不变性新猜想。
Conic locus of inversive Poncelet circumcenter and two points of invariant circle power
本文证明在通用庞加莱三角形族中，反演三角形的圆心轨迹为圆锥曲线，并验证了一个早期猜想：在通用庞加莱三角形中，存在两个独特的点，分别对三角形族的 circumcircle 和 Euler 圆保持恒定的圆幂。
The Brocard porism is a known 1d family of triangles inscribed in a circle and circumscribed about an ellipse. Remarkably, the Brocard angle is invariant and the Brocard points are stationary at the foci of the ellipse. In this paper we show that a certain derived triangle spawns off a second, smaller, Brocard porism so that repeating this calculation produces an infinite, converging sequence of porisms. We also show that this sequence is embedded in a continuous family of porisms.
The stationary focus of the Kiepert parabola over a special Poncelet triangle family
本文研究了在包含等边三角形的圆内接Poncelet三角形家族中，Kiepert内抛物线的焦点保持稳定性的现象。
Poncelet pairs of a circle and parabolas from a confocal family and Painlevé VI equations
研究圆与抛物线的Poncelet对，揭示等周期性性质，并构造Painlevé VI方程的代数解。
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 envelop rather elementary loci such as lines, circles, or points. Most phenomena are unproven though supported by solid numerical evidence (proofs are welcome). Some yet unrealized experiments are posed as "challenges" (results are welcome!).
We study center power with respect to circles derived from Poncelet 3-periodics (triangles) in a generic pair of ellipses as well as loci of their triangle centers. We show that (i) for any concentric pair, the power of the center with respect to either circumcircle or Euler's circle is invariant, and (ii) if a triangle center of a 3-periodic in a generic nested pair is a fixed affine combination of barycenter and circumcenter, its locus over the family is an ellipse.
Poncelet triangles: conic loci of the orthocenter and of the isogonal conjugate of a fixed point
本文研究了Poncelet三角形族中垂心与固定点等角共轭的轨迹，证明两者均为圆锥曲线，并揭示了特定条件下的抛物线、直线及圆分量特性。
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 the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii)…
We propose 3D generalizations of the Feuerbach theorem: the first one deals with a tetrahedron analogue of the Euler circle, the second one is done by means of an «up-in-ex-touch» construction. Then we give a geometric proof of the Grace theorem (a classical, but still not well-known, 3D Feuerbach theorem) and show its relation to the Poncelet closure theorem. Our elementary proof is based on the properties of imaginary generators on a sphere and of isotropic tangents to a conic. Then we show that the Grace theorem implies the Laguerre theorem, which generalises the Euler-Chapple formulas. Fur…
Poncelet Plectra: Harmonious Curves in Cosine Space
本文研究共焦椭圆台球中 Poncelet 三角形及其仿射像的余弦不变量，证明二者的余弦三元组扫过相同的拨片形三次曲线或球面曲线。
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 circles, ellipses, and teardrop-like curves. We derive expressions for both curves and their areas. We also study the locus of the vertices of Brocard triangles over the homothetic and Brocard-poristic Poncelet families.
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 circular caustic. Previously, determining if a locus was a conic was done on a case-by-case basis. In the confocal case, we also derive conditions under which a locus degenerates to a segment or a circle. We show the locus' turning number is +/- 3, while predicting its monotonicity with respect to the motion of a vertex of the triangle family.
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