Related papers: Generalized Chapple-Euler Relation

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes…

A convex quadrilateral, $Q$, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of $Q$ coincides with the midpoint of at least one of the diagonals of $Q$. A parallelogram, P, is a special case of a…

Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that…

For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's…

A certain real number, depending on two neighbouring sides of a quadrilateral and the diagonal meeting these two sides at their common point, is shown to be invariant under affinity. As an application we demonstrate a nice formula for the…

In this paper, we consider a time-periodically forced Kepler problem in any dimensions, with an external force which we only assume to be regular in a neighborhood of the attractive center. We prove that there exist infinitely many periodic…

Let $P$ be a finite point set in the plane. A \emph{$c$-ordinary triangle} in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of…

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted…

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…

In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathscr{A},\mathscr{B})$, with $\mathscr{A}$ smooth or singular and $\mathscr{B}$…

An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field $\mathbb{F}_q$ with characteristic greater than $3$,…

We generalize the classic definition of Delaunay triangulation and prove that for a locally finite and coarsely dense generic point set, $A \subseteq \mathbb{R}^d$, the $d$-simplices whose vertices belong to $A$ and whose circumscribed…

If $ABC$ is a given triangle in the plane, $P$ is any point not on the extended sides of $ABC$ or its anticomplementary triangle, $Q$ is the complement of the isotomic conjugate of $P$ with respect to $ABC$, $DEF$ is the cevian triangle of…

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…

Let $H$ be obtained from a cyclically $4$-edge-connected cubic planar graph $Y$ other than $K_4$ by deleting two adjacent vertices. We provide a short proof that if $H$ has circumference at least $k$ for some even integer $k \ge 4$, then…

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves, when $g\geq 3$ and the image of the $\ell$-adic cyclotomic character $G_k\to \Z^\times$ is infinite. Furthermore, we…

We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…

Among a triangle's exparabolas (parabolas escribed to the triangle), three are distinguished by having locally maximal parameter. They are determined by a simple cubic equation and characterized by having axes that contain the triangle's…

In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an…

We strongly believe that in order to prove two important geometrical pro\-blems in convexity, namely, the G. Bianchi and P. Gruber's Conjecture \cite{bigru} and the J. A. Barker and D. G. Larman's Conjecture \cite{Barker}, it is necessary…