Related papers: A Six-Point Ceva-Menelaus Theorem
We prove a generalization of the well known Routh's triangle theorem. As a consequence, we get a unification of the theorems of Ceva and Menelaus. A connection to Feynman's triangle is also given.
We generalize the classical Ceva's and Menelaus's theorems to curvilinear triangles bounded by circular arcs. We introduce trilinear coordinates associated with such triangles and develop several geometric constructions. In particular, for…
In the paper we prove generalization of Schl\"omilch's and Zetel's theorems about concurrent lines in a triangle. This generalization is obtained as a corollary of sharp geometric inequality about the ratio of triangular areas which is…
Suppose there are $n$ harmonic pencils of lines given in the plane. We are interested in the question whether certain triples of these lines are concurrent or if triples of intersection points of these lines are collinear, provided that we…
In the present paper, we prove generalizations of Banach, Kannan, Chatterjea, \'Ciri\'c-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mappings contracting perimeters of triangles. We consider corresponding…
John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present…
In this article we study Ceva's theorem and its higher-dimensional extensions from the perspective of algebraic and projective geometry. First, we situate the theorem within the study of algebraic surfaces by relating it to the defining…
In this article we prove a theorem that will generalize the concurrence theorems that are leading to the Franke's point, Kariya's point, and to other remarkable points from the triangle geometry.
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…
After having investigated and defined the ``surface of a translation-like triangle" in each non-constant curvature Thurston geometry \cite{Cs-Sz25}, we generalize the famous Menelaus' and Ceva's theorems for translation triangles in the…
In this paper we deal with $\NIL$ geometry, which is one of the homogeneous Thurston 3-geometries. We define the "surface of a geodesic triangle" using generalized Apollonius surfaces. Moreover, we show that the "lines" on the surface of a…
Lie symmetries of various geometrical and physical quantities in general relativity play an important role in understanding the curvature structure of manifolds. The Riemann curvature and Weyl tensors are two fourth-rank tensors in the…
We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a…
Semilinear clannish algebras have been recently introduced by the first author and Crawley-Boevey as a generalization of Crawley-Boevey's clannish algebras. In the present paper, we associate semilinear clannish algebras to the (colored)…
We note an intimate connection between the Lefschetz Theorem for c-arrangements, and a theorem of Hironaka relating the complement of an arrangement to its boundary manifold. This results in a generalization of Hironaka's result.
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…
Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the…
The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated which includes the case of converging curves with endpoints and the case in which the limit points assigned since the…
A set of $n$ points in the Euclidean plane determines at least $n$ distinct lines unless these $n$ points are collinear. In 2006, Chen and Chv\'atal asked whether the same statement holds true in general metric spaces, where the line…
It is shown in our earlier paper that, using only tools of elementary geometry, the classical Routh's theorem for triangles can be fully extended to tetrahedra. In this article we first give another proof of Routh's theorem for tetrahedra…