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We give necessary and sufficient conditions, both algebraic and geometric, for a quadrilateral to be the level set of the sum of the distances to m ≥ 2 different lines.

Linear and Multilinear Algebra, 70 (20), ISSN:1026-7573, ISSN:0308-1087, ISSN:1563-5139

In geometry, a Heron triangle is a triangle with rational side lengths and integral area. We investigate Heron triangles and their elliptic curves. In particular, we provide some new results concerning Heron triangles and give elementary proofs for some results concerning Heronian elliptic curves.

Given a cubic K. Then for each point P there is a conic $$C_P$$ associated to P. The conic $$C_P$$ is called the polar conic of K with respect to the pole P. We investigate the situation when two conics $$C_0$$ and $$C_1$$ are polar conics of K with respect to some poles $$P_0$$ and $$P_1$$ , respectively. First we show that for any point Q on the line $$P_0P_1$$ , the polar conic $$C_Q$$ of K with respect to Q belongs to the linear pencil of $$C_0$$ and $$C_1$$ , and vice versa. Then we show that two given conics $$C_0$$ and $$C_1$$ can always be considered as polar conics of some cubic K with respect to some poles $$P_0$$ and $$P_1$$ . Moreover, we show that $$P_1$$ is determined by $$P_0$$ , but neither the cubic nor the point $$P_0$$ is determined by the conics $$C_0$$ and $$C_1$$ .

For a set M, fin(M) denotes the set of all finite subsets of M, M² denotes the Cartesian product M × M, [M]² denotes the set of all 2-element subsets of M, and seq¹⁻¹(M) denotes the set of all finite sequences without repetition which can be formed with elements of M. Furthermore, for a set S, let |S| denote the cardinality of S. Under the assumption that the four cardinalities , |[M]², |M|², |fin(M)|, |seq¹⁻¹(M)| are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF., Annals of Pure and Applied Logic, 174 (2), ISSN:0168-0072, ISSN:0003-4843

More than a century ago, Norman Anning conjectured that it is possible to arrange 48 points on a circle, such that all distances between the points are integer numbers and are all among the solutions of the diophantine equationx2 + xy + y2 = 72 132 192 312:We shall obtain Anning's conjecture as a consequence of a far more general geometrical result.Key words: Quadratic diophantine equations, quadratic forms, plane integral point sets.

Linear Algebra and its Applications, 609, ISSN:0024-3795, ISSN:1873-1856

We give a projective proof of the butterfly porism for cyclic quadrilaterals and present a general reversion porism for polygons with an arbitrary number of vertices on a conic. We also investigate projective properties of the porisms., Comment: 12 figures

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter's construction we provide a new parametrisation of elliptic curves with torsion group Z/2ZxZ/8Z and give some configurations with all their points on a cubic curve., Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry, 63 (4), ISSN:0138-4821, ISSN:2191-0383

A pair (a,b) of positive integers is a pythagorean pair if a2+b2=□ (i.e., a2+b2 is a square). A pythagorean pair (a,b) is called a double-pythapotent pair if there is another pythagorean pair (k,l) such that (ak,bl) is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair (k,l) which is not a multiple of (a,b), such that (a2k,b2l) is a pythagorean pair. To each pythagorean pair (a,b) we assign an elliptic curve Γa,b with torsion group Z/2Z×Z/4Z, such that Γa,b has positive rank over Q if and only if (a,b) is a double-pythapotent pair. Similarly, to each pythagorean pair (a,b) we assign an elliptic curve Γa with torsion group Z/2Z×Z/8Z, such that Γa has positive rank over Q if and only if (a,b) is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve Γ with torsion group Z/2Z×Z/8Z is isomorphic to a curve of the form Γa, where (a,b) is a pythagorean pair. As a side-result we get that if (a,b) is a double-pythapotent pair, then there are infinitely many pythagorean pairs (k,l), not multiples of each other, such that (ak,bl) is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs., Journal of Number Theory, 233, ISSN:0022-314X, ISSN:1096-1658

In diesem Kapitel wollen wir die bisher hergeleitete Theorie anwenden, umein Apollonisches Beruhrungsproblem zu losen. Bei dem ausgesuchten Problem geht es darum, zu drei gegebenen Kreisen einen vierten Kreis zu konstruieren, welcher jeden der drei gegebenen Kreise beruhrt.

In diesem Kapitel betrachten wir eine Abbildung der Ebene auf sich selbst, die sogenannte Inversion am Kreis, und untersuchen geometrische Eigenschaften dieser Abbildung. Insbesondere werden wir sehen, dass diese Abbildung Kreise und Geraden auf Kreise und Geraden abbildet, aber nicht unbedingt Kreise in Kreise und Geraden in Geraden.

The exponential pencil Gλ:=G1(G−10G1)λ−1, generated by two conics G0,G1, carries a rich geometric structure: It is closed under conjugation, it is compatible with duality and projective mappings, it is convergent for λ→±∞ or periodic, and it is connected in various ways with the linear pencil gλ=λG1+(1−λ)G0. The structure of the exponential pencil can be used to characterize the position of G0 and G1 relative to each other., Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry, 59 (3), ISSN:0138-4821, ISSN:2191-0383

In Example 1.2 we gave a proof of 1 + 1 = 2 in 17 (!) proof steps. At that point you may have asked yourself: If it takes that much effort to prove such a simple statement, how can one ever prove any non-trivial mathematical result using formal proofs? This objection is of course justified; however, we will show in this chapter how one can simplify formal proofs using some methods of proof such as proofs by cases or by contradiction.

In 1931, Gӧdel proved his FIRST INCOMPLETENESS THEOREM which states that if PA is consistent, then it is incomplete, i.e.

Sometimes it is convenient to extend a given signature L by adding new non-logical symbols which have to be properly deffned within the language L or with respect to a given L-theory T.

In this section, we will show that ZF is suffciently strong to prove that PA is consistent.

The goal of this chapter is to develop the formal language of First-Order Logic from scratch. At the same time, we introduce some terminology of the so-called metalanguage, which is the language we use when we speak about the formal language (e.g., when we want to express that two strings of symbols are equal).

The key ingredient for Gӧdel’s Incompleteness Theorems is the so-called Gӧdelisation process which allows us to code terms, formulae and even proofs within PA.

By Gӧdel’s Completeness Theorem 5.5 we know that every consistent theory T has a model, and if T has an infinite model, then it also has arbitrarily large models.