Triangles II: Complex triangle coordinates

Summary This paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with complex triangle functions.     

Circle geometry in affine Cayley-Klein planes

Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular, we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles are parallel or have a point in common. For proving these statements, we use generalized complex numbers. Supported by a grant D01-761/24.10.06 from the Ministry of Education and Sciences, and by a grant 108/2007 from Sofia University.     

Triangles III: Complex triangle functions

Summary This paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In this paper, we look at the triangle coordinates of the special points of a triangle, and show that they are functions of its shape. We then show how these functions can be used to prove theorems about triangles, and to gain some insight into what makes a special point of a triangle a centre.     

Triangles I: Shapes

Summary This paper is the first in a series of three examining Euclidean triangle geometry via complex cross ratios. In this paper we show that every triangle can be characterized up to similarity by a single complex number, called its shape. We then use shapes and two basic theorems about shapes to prove theorems about similar triangles. The remaining papers in this series will examine complex triangle coordinates and complex triangle functions.     

Geometries of type [L.Af*]

Let be a [L.Af*]-geometry, that is a rank 3 geometry with linear spaces as plane residues, with dual affine planes as point residues and with generalized digons as line residues. Assume that (LL) holds in . In the particular case where the plane residues are finite circles, the structure of such geometries has been strongly restricted by A. P. Sprague. Moreover, C. Lefèvre and L. Van Nypelseer have given a complete classification of such geometries under the assumption that the plane residues are affine planes. We generalize these two results for [L.Af*]-geometries.Aspirant du Fonds National Belge de la Recherche Scientifique.     

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.     

Configurations of inscribed equilateral triangles

This paper explores equilateral triangles XYZ with vertices on sidelines of a given triangle ABC such that one side of XYZ is parallel to the corresponding side of ABC. There are six such triangles. They have many interesting properties which we investigate using trilinear coordinates. Our results improve and add to the earlier results of Blas Herrera Gómez about these configurations. We obtain new characterizations of several central points of the triangles and identify interesting pairs of triangles that are homologic (or perspective) and orthologic. The recognition of the Darboux cubic of a triangle is also accomplished in these configurations. Triples of circles intersecting in a point and six points on a conic also appear.      

On Napoleon's theorem in the isotropic plane]{On Napoleon's theorem in the isotropic plane

Summary Napoleon's original theorem refers to arbitrary triangles in the Euclidean plane. If equilateral triangles are externally erected on the sides of a given triangle, then their three corresponding circumcenters form an equilateral triangle. We present some analogous theorems and related statements for the isotropic (Galilean) plane.     

An extension of the Erdős-Szekeres theorem on large angles

The existence of a functionn(ε) (ε>0) is established such that given a finite setV in the plane there exists a subsetW⊆V, |W|<n(ε) with the property that for anyv εV\ W there are two pointsw ₁,w ₂ εW such that the angle ∢(w ₁ vw ₂)>π-ε.     

The Erdős-Mordell inequality is equivalent to non-positive curvature

We prove that, in plane absolute geometry, the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.      

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

On the lattice property of the plane and some problems of Dirac,Motzkin and Erdős in combinatorial geometry

LetS be a set ofn non-collinear points in the Euclidean plane. It will be shown here that for some point ofS the number ofconnecting lines through it exceedsc · n. This gives a partial solution to an old problem of Dirac and Motzkin. We also prove the following conjecture of Erdős: If any straight line contains at mostn−x points ofS, then the number of connecting lines determined byS is greater thanc · x · n. Dedicated to Paul Erdős on his seventieth birthday     

On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.     

On a problem of H. N. Gupta

It is shown that the axiom For any points x, y, z such that y is between x and z, there is a right triangle having x and z as endpoints of the hypotenuse and y as foot of the altitude to the hypotenuse, when added to three-dimensional Euclidean geometry over arbitrary ordered fields, is weaker than the axiom Every line which passes through the interior of a sphere intersects that sphere.     

Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ ₁(w), . . . , Δ _r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ _k (w)/Δ _k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.     

Angles between Euclidean subspaces

The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.Supported by the National Natural Science Fund of P.R. China.     

A completion problem for finite affine planes

A partial affine plane (PAP) of ordern is ann ²-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn ²+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn ²+n−√n can be completed and that forn≠3 any PAP withn ²+n−2 lines can be completed.     

Piecewise linear wavelets on sierpinski gasket type fractals

Let SG denote the Sierpinski gasket with Hausdorff measure μ of dimensionlog 3/log 2, let PL_k denote the continuous piecewise linear functions with respect to the usual triangulation of SG into 3^k triangles, and let W_k denote the orthogonal complement of PL_k−1 in PL_k. We construct a basis for each W_k, so that the entire collection is a frame for L²(dμ). This wavelet basis is obtained from three wavelet generators by scaling, translation and rotation, and the wavelets are supported either by one corner triangle or a pair of adjacent triangles in the triangulation of level k − 1. Analogous bases are constructed in the von Koch curve, the hexagasket, and the n-dimensional analog of SG.     

Remarks, du côté de chez Tarski, on symmetric ternary relations

We point out that the results in [12] are the model-theoretic counterpart of results established syntactically in [3] and [10], and that Martin’s theorem for the Euclidean 3- or higher-dimensional case, established in [5], does not depend on the Beckman-Quarles theorem, and can be rephrased as a result about axiomatizability and definability.