相关论文: Intriguing Invariants of Centers of Ellipse-Inscri…

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…

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 centers of inscribed circles in triangles, the 3-periodic orbits of an elliptic billiard, is also an ellipse. In this work we obtain the canonical equation of this ellipse, complementing the previous results obtained by O.…

Let R be a four-sided convex polygon in the xy plane and let M1 and M2 be the midpoints of the diagonals of R. It is well-known that if E is an ellipse inscribed in R, then the center of E must lie on Z, the open line segment connecting M1…

We consider 3-periodic orbits in an elliptic billiard. Numerical experiments conducted by Dan Reznik have shown that the locus of the centers of inscribed circles of the corresponding triangles is an ellipse. We prove this fact by the…

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…

A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle…

We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an…

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 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…

Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new "focus-inversive" family inscribed in Pascal's Lima\c{c}on. The following are some of its surprising invariants: (i)…

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 study triangles $ABC$ and points $P$ for which the generalized orthocenter $H$ corresponding to $P$ coincides with a vertex $A,B$, or $C$. The set of all such points $P$ is a union of three ellipses minus $6$ points. In addition, if…

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 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…

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…

Given a triangle, a trio of circumellipses can be defined, each centered on an excenter. Over the family of Poncelet 3-periodics (triangles) in a concentric ellipse pair (axis-aligned or not), the trio resembles a rotating propeller, where…

We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure…

Several authors have remarked the convenience of understanding the different notions of center appearing in Geometry (centroid of a set of points, incenter of a triangle, center of a conic and many others) as functions. The most general way…

We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed…