Related papers: Kasner meets Poncelet
We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its…
In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially…
This article proves the following theorem, first enunciated by Roger Penrose about 70 years ago but never published: In $\mathbb{R}P^{2}$, if conics are assigned to seven of the vertices of a combinatorial cube such that (i) conics…
In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald's constant term identities admit an extra set of free parameters, thereby…
Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex pentagons of the same area and the same perimeter.
This work studies circle-geometry methods through their application to a main theorem about circles tangent twice to a conic. The authors investigate the Sharygin point -- a point lying in the pencil of two non-intersecting circles -- and…
The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between…
The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in…
We study the Carnot theorem and the configuration of points and lines in connection with it. It is proven that certain significant points in the configuration lie on the same lines and same conics. The proof of an equivalent statement…
Main Theorem. Two parabols have four common points. There exists a circle tangent to the sides of the obtained parabolic quadrilateral if and only if the diagonals of this quadrilateral are orthogonal. The proof of the Main Theorem is…
Chasles' Quadrilateral Theorem is a classical statement about four tangents to a conic that simultaneously circumscribe a circle. In its various formulations, it relates the concurrence of certain lines to the existence of confocal conics…
The Newton line and the associated theorems by Newton and Gauss for tetragons and quadrilaterals are closely linked to some other theorems of Euclidean geometry: a theorem by Bocher on the existence of a nine-point conic of a quadrangle, a…
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
Two vertex-labelled polygons are \emph{compatible} if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations---for every face, the clockwise cyclic…
In 1640's, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a…
For a given triangle $\triangle ABC$, we define two sequences of points on line $BC$ and provide their generalizations to real functions such that centers of circumscribed circles around $A$ and adjacent points in subsequences generate a…
We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard…
We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we…
We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem…
We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an $O(n\log n)$-time algorithm for the two-center…