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We consider some applications of Ptolemy's theorem. In particular, we nd a criterion for constructing an inscribed hexagon.

There are many problems whose solution requires proof that a quadrilateral is cyclic. The main reason for writing this paper is to offer a number of new tools for proving that a particular quadrila...

Na početku rada prisjećamo se osnovne podjele četverokuta, temeljnih pojmova i činjenica potrebnih za ostatak rada. U drugom poglavlju bavimo se Eulerovim teoremom koji je poopćenje relacije paralelograma, dajemo formulu za površinu općeg konveksnog četverokuta i proširenje Teorema o kosinusu. Dajemo i poopćenje dobro poznatog Ptolomejevog teorema. U trećem poglavlju definiramo težište i težišnice četverokuta, dajemo iskaz i dokaz Varignonovog teorema i njegove posljedice, te se bavimo drugim osobitim točkama u četverokutu. At the beginning of the paper, we recall the basic classification of the quadrilateral, the basic concepts and facts needed for the rest of the paper. In the second chapter we deal with the Euler’s theorem which is a generalization of the relation of parallelograms, we give a formula for the surface of a general convex quadrilateral and an extension of the cosine theorem. We also give a generalization of the well-known Ptolemy’s theorem. In the third chapter, we define the centroids and the medians of the quadrilateral, give a statement and proof of Varignon’s theorem and its consequences, and deal with other significant points in the quadrilateral.

We give generalizations of Casey’s theorem and its converse for higher dimensions. We also present a multidimensional generalization for the problem of Apollonius. To do this we introduce a notion of ψ-tangent for a generalized k-sphere that touches a number of generalized n-balls in proper manner.

U ovom diplomskom radu bavimo se Caseyevim teoremom. U prvom poglavlju iskazujemo i dokazujemo Ptolomejev teorem kako bismo pobliže razumjeli Caseyev teorem, te definiramo osnovne pojmove koje koristimo kroz rad. U drugom poglavlju pažnju posvećujemo inverziji koja se koristi u dokazivanju Caseyevog teorema. U trećem poglavlju iskazujemo i dokazujemo Caseyev teorem na nekoliko načina i dokazujemo specijalne slučajeve Caseyevog teorema. Sami kraj rada posvećen je primjeni Caseyevog teorema u dokazivanju drugih teorema i u rješavanju geometrijskih zadataka. In this graduate thesis we are dealing with Casey’s theorem. In the first chapter, we formulate and prove Ptolemy’s theorem to better understand Casey’s theorem, and define the basic terms we use through graduate thesis. In the second chapter, we dedicate to the inversion used to prove Casey’s theorem. In the third chapter, we phrase and prove Casey’s theorem in several ways and prove the special cases of Casey’s theorem. The end of the graduate thesis is dedicated to the application of Casey’s theorem in proving other theorems and in solving geometric problems.

International audience; This book consists of a series of self-contained essays on non-Euclidean geometry in a broad sense, including the classical geometries of constant curvature (spherical and hyperbolic), de Sitter, anti-de Sitter, co-Euclidean, co-Minkowski geometries, Hermitian geometry, and some axiomatically defined geometries. Some of these essays deal with very classical questions and others address problems that are at the heart of present day research, but all of them are concerned with fundamental topics.

U ovom diplomskom radu proučavamo Ptolomejev teorem i njegove inačice. Prvo, prikazujemo nekoliko različitih dokaza Ptolomejeva teorema. Nakon toga, pomoću Ptolomejeva teorema dokazujemo neke klasične teoreme iz elementarne geometrije, kao i nekoliko složenijih geometrijskih problema. Konačno, dajemo i nekoliko poopćenja Ptolomejeva teorema. In this thesis we study Ptolemy's theorem and its variants. First we give several proof of Ptolemy's theorem. Then, we utilize the theorem for proving several classical theorems in elementary geometry. In addition, by virtue of Ptolemy's theorem we solve some complex geometric problems. Finally, we also study some generalizations of Ptolemy's theorem.

In this paper we characterize Ptolemy circles and Ptolemy segments up to isometry. Moreover, we present an example of a metric sphere, which is Mobius equivalent but not homothetic to the standard metric sphere with its chordal metric.

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ with $n$ chords is called an $n$-crossing if all chords of $E$ are mutually crossing. A chord diagram $E$ is called nonintersecting if $E$ contains no $2$-crossing. For a chord diagram $E$ having a $2$-crossing $S = \{ x_1 x_3, x_2 x_4 \}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1 = (E \setminus S) \cup \{ x_2 x_3, x_4 x_1 \}$ or $E_2 = (E \setminus S) \cup \{ x_1 x_2, x_3 x_4 \}$. It is shown that there is a one-to-one correspondence between the multiset of all nonintersecting chord diagrams generated from an $n$-crossing with a finite sequence of expansions and the set of alternating permutations of order $n+1$.

© 2016 Kostin, A.V., Kostina, N.N.We obtain interpretations of Casey's theorem and of its hyperbolic version in a pseudo-Euclidean and in a pseudo-hyperbolic spaces.

In this article we introduce the reader to some pretty results concerning a circle and regions within it defined by various line segments. We have framed the results in terms of a pizza, so we could well call this an introduction to Pizza Geometry! The novel aspect is the variety of approaches used to prove these results.

We provide a visual proof of Pythagorean theorem. The main idea of the proof is to compute (a + b)2 in two different ways: one with aid of Ptolemy's theorem and the other one by dissecting a square.

(2004). A Vector Approach to Ptolemy's Theorem. Mathematics Magazine: Vol. 77, No. 5, pp. 386-388.

Brahmagupta extended Ptolemy’s theorem on cyclic quadrilaterals to find the lengths of the diagonals, the segments made when they are cut at the point of intersection of the diagonals, and the lengths of the sides of the needles, the figures formed when opposite sides of the quadrilateral are extended until they meet. Proofs of these results are given, and a derivation of the 19th century result of the length of the third diagonal is given. This “diagonal” is formed by connecting the tips of the needles with a line segment.

Summary. The goal of this article is to formalize some theorems that are in the Freek Wiedijk’s list ”Formalizing 100 Theorems” on the web. These are elementary theorems included in every handbook of Euclidean geometry and trigonometry: the law of cosines, the Heron’s formula, the isosceles triangle theorem, the intersecting chords theorem and the Ptolemy’s theorem.

Ptolemy's equality for four points on a circle is related to a Feuerbach-type area relation. This suggested an extension of Ptolemy's inequality to a Feuerbach type volume relation between simplexes formed from n+2 points in Rn (n≥2). Extensions of the Mobius-Neuberg and Pompeiu Theorems in R2 are given for Rn. Ptolemy's inequality is also extended to convex n-gons in R2 yielding an extension of Fuhrmann's hexagon theorem to 2n-gons in R2 (n≥3).

(1992). Proof without Words: The law of cosines via Ptolemy's Theorem. Mathematics Magazine: Vol. 65, No. 2, pp. 103-103.

It might be both profitable and interesting to attempt to evaluate the relative importance of some of the outstanding theorems of elementary geometry by means of a poll of mathematicians. If this were done, there is little doubt that the theorem of Pythagoras would in the opinion of most people, head the list. It is not so clear just how other important theorems would be ranked. But without doubt, Ptolemy's theorem concerning an inscribed quadrilateral is deserving of a high place. This judgemnt is warranted, both from the point of view of the use which Ptolemy made of it, and from a consideration of its power as a tool for geometric study.