Perspective Graves triads and triads of perspective inscribed triangles

Any Pappus configuration can be regarded (in six ways) as a triad of triangles, each inscribed in the previous one; such a triad of triangles is called aGraves triad. Graves triads in which each pair of triangles is in perspective may be calledperspective Graves triads; such triads occur in connection with various results in both projective and metrical geometry. In this paper we give a synthetic proof of the existence of perspective Graves triads; the proof is of interest because it holds in allharmonic planes and requires neither the axiom of Pappus nor the full axiom of Desargues. The paper also includes synthetic proofs of other related results.(Results involving perspective Graves triads can be found in [l, p.149] and [2]. This paper is the result of recent correspondence with Guinand concerning [2] and [3].)     

Generalized double involutive geometry of the triangle

We study a generalized geometry of the triangle, based on the idea of letting two arbitrary points play the role that the centroid and the orthocenter play classically. We thus generalize some of the classical results and constructions and also prove some new results in ordinary metric geometry.     

Graves triads,Möbius pairs,and related matrices

A Graves triad is a cyclic triad of triangles, each circumscribing the next, forming a Pappus configuration. If two of the triangles are in perspective, then so are the other two pairs. In a non-cyclic triad of such circumscribing triangles, if two pairs are in perspective, then so is the third pair. These results correspond to certain properties of 3×3 zero-diagonal matrices. There is also an analogous result for Möbius pairs of mutually circumscribing tetrahedra and 4×4 zero-diagonal matrices.     

Functional equations associated with triangle geometry

Summary Triangle geometry is treated in the context of functional equations of three variablesa, b, c which may be regarded as the sidelengths of a variable triangle. Trianglecenters (e.g., incenter, circumcenter, centroid), andcentral lines (e.g., the Euler line) are defined and partitioned into classes:0-centers, 1-centers, 2-centers and0-lines, 1-lines, and 2-lines. Criteria for parallelism, perpendicularity, and other geometric relations are proved in terms of these classes. The Euler line and central lines parallel or perpendicular to the Euler line serve as examples.     

Paolo Pizzetti: The forgotten originator of triangle comparison geometry

The theorem comparing the angles of two geodesic triangles with the same side lengths lying on surfaces with different curvatures, commonly attributed for the two-dimensional case to A.D. Alexandrov (1948) and for the n-dimensional Riemannian case to V.A. Toponogov (1959), goes back, for the two-dimensional case, to Paolo Pizzetti (1907b). Besides suggesting a correction of the historical narrative regarding the development of comparison geometry, the present note also mentions possible reasons for the oversight.     

The median triangle theorem as an entrance to certain issues in higher-dimensional geometry

Mathematische Semesterberichte - The median triangle theorem states that the three medians of a triangle can serve as the sides of another triangle. This theorem together with other...     

In search of more triangle centres. A source of classroom projects in Euclidean geometry

A point E inside a triangle ABC can be coordinatized by the areas of the triangles EBC, ECA, and EAB. These are called the barycentric coordinates of E. It can also be coordinatized using the six segments into which the cevians through E divide the sides of ABC, or the six angles into which the cevians through E divide the angles of ABC, or the six triangles into which the cevians through E divide ABC, etc. This article introduces several coordinate systems of these types, and investigates those centres of ABC whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of ABC, such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry.     

On a generalization of duality triads

Some aspects of duality triads introduced recently are discussed. In particular, the general solution for the triad polynomials is given. Furthermore, a generalization of the notion of duality triad is proposed and some simple properties of these generalized duality triads are derived.     

Problematizing the didactic triangle

This paper suggests that the typical framing of the didactic triangle in the English-speaking literature is narrower than it should be, and that it should be broadened to view classroom activities from a more social/cultural perspective. A framework for examining mathematically productive classrooms is given and exemplified, with the exemplification providing a point of departure for the kinds of analytic work that might be done.     

Reflecting a triangle in the plane

We prove that if the three angles of a triangleT in the plane are different from (60°, 60°, 60°), (30°, 30°, 120°), (45°,45°,90°),(30°,60°,90°), then the set of vertices of those triangles which are obtained fromT by repeating ‘edge-reflection’ is everywhere dense in the plane.     

Several identities in the Catalan triangle

In this paper, we first establish several identities for the alternating sums in the Catalan triangle whose (n, p) entry is defined by B _{n, p} = $ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) $ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) . Second, we show that the Catalan triangle matrix C can be factorized by C = FY = ZF, where F is the Fibonacci matrix. From these formulas, some interesting identities involving B _{n, p} and the Fibonacci numbers F _n are given. As special cases, some new relationships between the well-known Catalan numbers C _n and the Fibonacci numbers are obtained, for example:

Cubic curves in the triangle plane

The subject of this paper are two pencils of cubic curves that are the result of certain geometrical constructions in the triangle plane. One of them turns out to be the probably most significant pencil of anallagmatic cubics that are associated with triangle geometry. Both contain virtually all important single cubics, and other well known curves appear closely connected with them.Dedicated to Professor T.G. Ostrom on the occasion of his 80th birthday     

New identities in the Catalan triangle

In this paper we prove new identities in the Catalan triangle whose (n,p) entry is defined by

     

On the type of triangle groups

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.