Theorems Of Euclidean Geometry Research Articles

Theorems of Euclidean Geometry Through Calculus

We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector formulae, by assuming the existence of an unknown relation between the geometric quantities at stake, observing how the relation behaves under small deviations of those quantities, and naturally establishing differential equations that we integrate out. Applying the general solution to some specific situation gives a particular solution corresponding to the expected theorem. We also establish an equivalence between a polynomial equation and a set of partial differential equations. We finally comment on a differential equation which arises after a small scale transformation and should concern all relations between metric quantities.

Guided discovery of the nine-point circle theorem and its proof

ABSTRACTThe nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.

On the Compatibility between Euclidean Geometry and Hume's Denial of Infinite Divisibility

It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is the distinction he draws between a precise and an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true.

Pratt sequences and n-gons

A Pratt sequence for an n-gon consists of n distinct lines, one through each vertex of the n-gon, which satisfy a certain cyclic product relation. If n=3, three lines through the vertices of a triangle form a Pratt sequence if and only if they are concurrent. The purpose of this paper is to generalise many familiar theorems of Euclidean geometry. We show that whenever three lines (such as the medians, angle-bisectors, altitudes, etc.) are concurrent in a triangle, then the corresponding n lines in an n-gon form a Pratt sequence.

Creating a Learning Environment for Geometric Reasoning

Geometry teachers can introduce the theorems of Euclidean geometry in various ways. Complementing the standbys of the past, namely, compass-and-straightedge constructions and manipulatives, are geometry-construction software, spreadsheets, and programmable graphing calculators.

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.

Automated production of traditional proofs for theorems in Euclidean geometry I. The Hilbert intersection point theorems

We present a method which can produce traditional proofs for a class of constructive geometry statements in Euclidean geometry. The method is a mechanization of the traditional area method used by many geometers. The key idea of our method is to eliminate dependent (constructed) points in a geometry statement using a few basic geometry propositions about the area of triangles. The method has been implemented. Our program, calledEuclid, can produce traditional proofs of many hard geometry theorems such as Pappus' theorem, Pascal's theorem, Gauss point theorem, and the Pascal conic theorem. Currently, it has produced proofs of 110 nontrivial theoremsentirely automatically. The proofs produced byEuclid are elegant, short (often shorter than the proofs given by geometers) and understandable even to high school students. This method seems to be the first that can produce traditional proofs for hard geometry theorems automatically.

Some examples of the use of distances as coordinates for Euclidean geometry

Distance geometry provides us with an implicit characterization of the Euclidean metric in terms of a system of polynomial equations and inequalities. With the aid of computer algebra programs, these equations and inequalities in turn provide us with a coordinate-free approach to proving theorems in Euclidean geometry analytically. This paper contains a brief summary of the mathematical results on which this approach is based, together with some examples showing how it is applied. In particular, we show how it can be used to derive the topological structure of a simple linkage mechanism.

Fixed Point Theorems in Euclidean Geometry

The direction of future high school geometry courses is currently the subject of much discussion. One frequent suggestion is that high school geometry should be presented with transformation theory as the unifying theme. In support of this new direction, we shall illustrate that transformations can be employed to bring theorems from classical synthetic geometry into the so-called mainstream of modern mathematics. The thread tying these two points of view together will be the application of fixed point theorems.

Graphical Determination of Two-Port Properties by the Bilinear Pole-Zero Method,

By locating the principle pole and zero of the relevant bilinear function and utilizing some lesser known theorems in Euclidean geometry a simple method is outlined for the evaluation of common design formulas for active bilateral two-port networks. The method is particularly suitable where simultaneous criteria involving power gain, voltage gain or other transfer ratios, stability or impedance level, must be satisfied in the presence of a variable parameter. Various transformations and their merits are outlined, one of the most important being a method for employing the Smith chart in the above evaluations. The relationship of this method to known Euclidean graphical procedures, e.g., the image circle and the Linvill chart is given, indicating these to be special cases within the more general theory.

The fundamental theorems of elementary geometry. An axiomatic analysis

Introduction. The present investigation is concerned with an axiomatic analysis of the four fundamental theorems of Euclidean geometry which assert that each of the following triplets of lines connected with a triangle is copunctual: the medians, the altitudes, the perpendicular bisectors, and the bisectors of the angles. The general framework for our discussion will be provided by an affine plane (which is obtained from a projective plane by deleting a line and its points). But in order to enunciate these theorems we have to add to the concepts provided by affine geometry two further relations: the relation of the midpoint and that of orthogonality. There exists one important difference between these two relations. If one subjects the midpoint relation to some obvious and formal restrictions, then there exists at most one such relation. But there exist always different orthogonality relations, since affine transformations transform one such relation into a different one; and even if one considers such orthogonality relations as not essentially different and considers only orthogonality relations meeting quite a fair amount of requirements, then uniqueness will be an exceptional case. The existence of a midpoint relation is equivalent to the closure of certain configurations, to the existence of sufficiently many reflections in points, and to the following algebraic criterion: the plane under consideration is the plane over a right distributive Cartesian number system of characteristic different from 2 (using a concept introduced in an earlier paper(')). If the characteristic is 3, then the medians of a triangle are parallel, otherwise they are copunctual. Given a midpoint relation and an orthogonality relation meeting the obvious and formal requirements, then the theorem of the altitudes and that of the perpendicular bisectors of a triangle are equivalent; and they are both equivalent to the fact that the plane under consideration is the plane over an ordinary commutative field of characteristic different from 2 and that orthogonality may be defined in terms of a quadratic form y2 -CX2, c O0. The relation between these theorems and the theorem of the bisectors of the angles is not as clearcut. For, the latter theorem is a consequence of the former ones; but the converse can be obtained only by adding two further statements concerning the existence of bisectors of angles, which does not seem to be assured by

On mathematical concepts of the material world

The object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the Material World. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the “stuff” in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows:—Given a set of entities which form the field of a certain polyadic ( i. e ., many-termed) relation R . What “axioms” satisfied by R have as their consequence that the theorems of Euclidean Geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean Space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean Geometry. The solution of this narrower problem of the axioms of geometry is assumed ( cf . Part II, Concept I) without proof in the form most convenient for this wider investigation. Poincaré has used language which might imply the belief that, with the proper definitions, Euclidean Geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem here discussed is to find various formulations of axioms concerning R , from which, with appropriate definitions, the Euclidean Geometry issues as expressing properties of the field of R . In view of the existence of change in the Material World, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration.

XIV. On mathematical concepts of the material world

The object of this memoir is to initiate the mathematical investigation of various possible ways of conceiving the nature of the material world. In so far as its results are worked out in precise mathematical detail, the memoir is concerned with the possible relations to space of the ultimate entities which (in ordinary language) constitute the “stuff” in space. An abstract logical statement of this limited problem, in the form in which it is here conceived, is as follows: Given a set of entities which form the field of a certain polyadic ( i. e ., many-termed) relation R, what “axioms” satisfied by R have as their consequence, that the theorems of Euclidean geometry are the expression of certain properties of the field of R ? If the set of entities are themselves to be the set of points of the Euclidean space, the problem, thus set, narrows itself down to the problem of the axioms of Euclidean geometry. The solution of this narrower problem of the axioms of geometry is, assumed ( cf . Part II., Concept I.) without proof in the form most convenient for this wider investigation. But in Concepts III., IV., and V., the entities forming the field of R are the “stuff,” or part of the “stuff,” constituting the moving material world. Poincaré has used language which might imply the belief that, with the proper definitions, Euclidean geometry can be applied to express properties of the field of any polyadic relation whatever. His context, however, suggests that his thesis is, that in a certain sense (obvious to mathematicians) the Euclidean and certain other geometries are interchangeable, so that, if one can be applied, then each of the others can also be applied. Be that as it may, the problem, here discussed, is to find various formulations of axioms concerning R, from which, with appropriate definitions, the Euclidean geometry issues as expressing properties of the field of R. In view of the existence of change in the material world, the investigation has to be so conducted as to introduce, in its abstract form, the idea of time, and to provide for the definition of velocity and acceleration. The general problem is here discussed purely for the sake of its logical ( i. e ., mathematical) interest. It has an indirect bearing on philosophy by disentangling the essentials of the idea of a material world from the accidents of one particular concept. The problem might, in the future, have a direct bearing upon physical science if a concept widely different from the prevailing concept could be elaborated, which allowed of a simpler enunciation of physical laws. But in physical research so much depends upon a trained imaginative intuition, that it seems most unlikely that existing physicists would, in general, gain any advantage from deserting familiar habits of thought.