Related papers: Ceva's triangle inequalities
We give a formula for counting the triangles in a picture consisting of the three sides of a triangle and some cevians. This lets us prove statements that are claimed without proof in the Online Encyclopedia of Integer Sequences and some…
For a given triangle $T$ and a real number $\rho$ we define Ceva's triangle $\CT_\rho(T)$ to be the triangle formed by three cevians each joining a vertex of $T$ to the point which divides the opposite side in the ratio $\rho:(1-\rho)$. We…
We establish a family of parametric isoperimetric-type inequalities with multiple geometric quantities for closed convex curves. These inequalities hold under certain parameter conditions. We also prove the equality conditions. Some new…
Given a plane triangle $\Delta$, one can construct a new triangle $\Delta'$ whose vertices are intersections of two cevian triples of $\Delta$. We extend the family of operators $\Delta\mapsto\Delta'$ by complexifying the defining two…
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…
We study the triangle inequalities for angles (with different definitions) and present inequalities concerning the entries of correlation matrices through the positivity of $3\times 3$ matrices. We extend our discussions to the inequalities…
We suggest a method of solving the problem of existence of a triangle with prescribed two bisectors and one third element which can be taken as one of the angles, the sides, the heights or the medians, or the third bisector.
We characterize quasiconformal mappings in terms of the distortion of the vertices of equilateral triangles.
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…
Three generalizations of the well-known Ceva's Theorem are given in this paper and some applications.
A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a particular polynomial in four real indeterminates.…
In 1840 Jacob Steiner on Christian Rudolf's request proved that a triangle with two equal bisectors is isosceles. But what about changing the bisectors to cevians? Cevian is any line segment in a triangle with one endpoint on a vertex of…
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…
If $P$ is a point inside $\triangle ABC$, then the cevians through $P$ divide $\triangle ABC$ into smaller triangles of various sizes. We give theorems about the relationship between the radii of certain excircles of some of these…
We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the…
There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already…
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…
If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle…
We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying…
We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the…