A journey round the triangle Lester’s circle, Kiepert’s hyperbola and a configuration from Morley

2003 ◽  
Vol 87 (509) ◽  
pp. 217-229 ◽  
Author(s):  
H. Martyn Cundy
Keyword(s):  
Simple Proof ◽  
Recent Article ◽  
Computer Assisted ◽  
Complex Numbers ◽  
Crucial Point ◽  
Rectangular Hyperbola ◽  
Equilateral Triangles ◽  
Point Of Intersection ◽  
Trilinear Coordinates ◽  
Interesting Theorem

In a recent article [1], Ron Shail has given a Cartesian proof of an interesting theorem due to J. A. Lester. This states that, for any triangle, the circumcentre O, the nine-point centre O9 and the two Fermat points F and Fʹ, (which are the points of concurrence of the joins of its vertices to the vertices of equilateral triangles drawn outwards/inwards on the opposite sides), are concyclic. He refers to Lester's own treatment as needing complex coordinates with computer-assisted algebra; his own proof uses an unpromising method, and results in similar problems. Contemplation of the configuration would suggest that the location of the point of intersection of FFʹ with the Euler line OO9 might lead to a simple proof. The theorem is in fact a corollary from the properties of a remarkable configuration originating with Morley [2, p. 209], and shown in Figure 1. He did not deduce Lester’s result, nor label the crucial point J in the diagram, which was drawn without that particular intersection. Also involved in this figure is a rectangular hyperbola known by the name of its describer Kiepert [3]. It is helpful to discuss all three together. What follows is a journey through country nowadays rather unfamiliar, avoiding the computerised motorway and using older tracks via complex numbers, trilinear coordinates and Euclidean methods which reveal much more than is apparent from a Cartesian treatment.

A simple proof of Lester’s theorem

2003 ◽  
Vol 87 (510) ◽  
pp. 444-452
Author(s):  
John Rigby
Keyword(s):  
Computer Algebra ◽  
Simple Proof ◽  
Computer Algebra System ◽  
Recent Article ◽  
Diagonal Form ◽  
Computer Assisted ◽  
Polynomial Equations ◽  
Scalene Triangle ◽  
Systems Of Polynomial Equations ◽  
Special Case

Lester’s theorem (1997) states that in any scalene triangle the two Fermat points F and F' (to be defined later), the nine-point centre N, and the circumcentre O, are concyclic, and that the pair of points O,F separates the pair N, F'. (In certain geometrical situations a line is regarded as a circle of infinite radius, so that the word ‘concyclic’ includes ‘collinear’ as a special case, but here ‘concyclic’ means ‘lying on a proper circle of finite radius’.) Previous proofs of Lester’s theorem have involved advanced techniques and/or computer algebra; to quote from Ron Shail’s recent article [1],‘Lester’s original computer-assisted discovery and proof make use of her theory of “complex triangle coordinates” and “complex triangle functions”. ... A proof has also been given by Trott ... using the advanced concept of GrObner bases in the reduction of systems of polynomial equations to “diagonal” form. Trott’s work uses the computer algebra system Mathematica as an essential tool.’

scholarly journals BioFilm©

2000 ◽  
Vol 8 (2) ◽  
pp. 6-7
Author(s):  
Lee van Hook
Keyword(s):  
Recent Article ◽  
Metal Halide ◽  
Toxic Effects ◽  
Polymer Science ◽  
Complex Combination ◽  
Dry Biomass ◽  
Equilateral Triangles ◽  
Silver Grains

Photographic chemistry has long been a complex combination of inorganic metal-halide and organic chemistries and polymer science. We at the P.R.I, have managed to add biology to this stew.Silver has long been known as a toxicelement to microbes, and so used as a drug to kill bacteria. But there are bacteria that can survive in environments high in silver. It has been reported that some bacteria can accumulate up to 25% of their dry biomass as silver, and so acquire resistance to the toxic effects of silver. Also, a recent article in the Proc. Nat. Acad.Sci. describes the intracellular deposition of silver grains in such shapes as hexagons and equilateral triangles.

scholarly journals Simple proof of a theorem on permanents

1969 ◽  
Vol 10 (1) ◽  
pp. 52-54 ◽  
Author(s):  
D. Ž. Djoković
Keyword(s):  
Simple Proof ◽  
Complex Matrix ◽  
Image Position ◽  
Interesting Theorem

Let A = (aij) be an n × n complex matrix. The permanent of this matrix iswhere the sum is taken over all permutations p of the set {1, …, n}.In a recent paper [1] E. H. Lieb proved an interesting theorem (see below) which he applied to verify some conjectures of M. Marcus and M. Newman. The purpose of this note is to give a simple proof of Lieb's theorem.

scholarly journals Equivalence Levels of Literary Corpus Translation Using a Freeware Analysis Toolkit

2021 ◽  
Vol 27 (1) ◽  
pp. 55-70
Author(s):  
Frans Sayogie ◽  
Moh. Supardi
Keyword(s):  
Machine Translation ◽  
Harry Potter ◽  
Literary Translation ◽  
Computer Assisted ◽  
Crucial Point ◽  
Creative Project ◽  
Word Level ◽  
Translation Equivalent ◽  
The Phoenix

Machine translation has the potential to make huge contributions to translation industries, but it seems, for now, that machine translation equivalence has led to a crucial point for literary translation by using machine translation because of the problem of the equivalence itself. This paper, therefore, aimed to see the equivalence degree of literary translation resulted by machine translation, i.e., freeware toolkit AntConc 3.5.0 software 2019. The data were collected from English-Indonesian J.K. Rowling's Harry Potter and The Order of The Phoenix novel by using the software to find the equivalent translation. The collected data were analyzed qualitatively based on the strategy of translation equivalent level proposed by Mona Baker (1992). The analyzed data revealed that the equivalent level of the software mostly occurred in a word level, above word level, and grammatical level. The software was likely difficult to find a textual level and a pragmatic level of translation equivalence because they required a context and still needed human involvement as part of a greater creative project of translation which were not done by the machine translation. After all, Antconc 3.5.0 as Computer Assisted Tool (CAT) brought a huge contribution to translation industry, helped to analyze large corpora particularly to find the degree of translation equivalence in word and above word level.

Plane polygons revisited

1982 ◽  
Vol 19 (A) ◽  
pp. 113-122 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Electrical Engineers ◽  
Equilateral Triangles ◽  
Arbitrary Plane ◽  
Arbitrary Triangle ◽  
Napoleon’S Theorem ◽  
Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

scholarly journals Problem:—To find a point M external to triangle ABC such that lines from it to the vertices A, B and C make equal angles with the sides, that is MCA1 = MAB = MBC

1913 ◽  
Vol 13 ◽  
pp. 156-157
Author(s):  
William Finlayson
Keyword(s):  
Rectangular Hyperbola ◽  
Point Of Intersection

It is evident that if MCA1 = MBC the point M lies on the circle O, which touches AC at C and passes through B. Again, if we take two equal and opposite pencils A(BIMC) and C(A1EMb) we know that their intersections lie on an hyperbola, and observe that when AI is the bisector of A its corresponding ray CE of the second pencil is parallel to it. This gives the direction of the asymptotes; the second asymptote will be parallel to the external bisector of A. The conic is therefore a rectangular hyperbola. When the ray of the first pencil coincides with AC1 the ray of the second corresponding to it will make with CA an angle equal to A, it is therefore parallel to AB, and the point of intersection is C; hence the second ray ab is a tangent at C, similarly BA is a tangent at A; therefore AC passes through the centre of the conic, and B′ is the centre. The point M where this conic cuts O is the required point M. Denote this hyperbola by γ, then on repeating this construction at B we get a second hyperbola β cutting the circle O1 at M1, which makes angle M1BA2 = M1AC = M1CB. (See Fig. on p. 157.)

An upper bound for the breadth of the Jones polynomial

1988 ◽  
Vol 103 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Morwen B. Thistlethwaite
Keyword(s):  
Simple Proof ◽  
Upper Bound ◽  
Recent Article ◽  
Jones Polynomial ◽  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Kauffman Bracket ◽  
Bracket Polynomial ◽  
Connected Diagram

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial 〈D〉 ∈ Z[A, A−1 of a link diagram D.

scholarly journals On Weyl fractional integral operators

2004 ◽  
Vol 35 (2) ◽  
pp. 169-174 ◽  
Author(s):  
Rashmi Jain ◽  
M. A. Pathan
Keyword(s):  
Hypergeometric Series ◽  
Fractional Integral ◽  
Integral Operators ◽  
Complex Numbers ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Arbitrary Complex ◽  
The Laplace Transform ◽  
Weyl Fractional Integral ◽  
Interesting Theorem

In this paper, we first establish an interesting theorem exhibiting a relationship existing between the Laplace transform and Weyl fractional integral operator of related functions. This theorem is sufficiently general in nature as it contains $n$ series involving arbitrary complex numbers $ \Omega(r_1,\ldots r_n) $. We have obtained here as applications of the theorem, the Weyl fractional integral operators of Kamp'e de F'eriet function, Appell's functions $ F_1 $, $ F_4 $, Humbert's function $ \Psi_1$ and Lauricella's, triple hypergeometric series $ F_E $. References of known results which follow as special cases of our theorem are also cited. Finally, we obtain some transformations of $ F^{(3)}$ and Kamp'e de F'eriet function with the application of our main theorem .

Plane polygons revisited

1982 ◽  
Vol 19 (A) ◽  
pp. 113-122 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Electrical Engineers ◽  
Equilateral Triangles ◽  
Arbitrary Plane ◽  
Arbitrary Triangle ◽  
Napoleon’S Theorem ◽  
Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

3-D organization of fibrinogen receptors using computer reconstruction and whole-mount microscopy

1988 ◽  
Vol 46 ◽  
pp. 30-31
Author(s):  
E. T. O'Toole ◽  
R. R. Hantgan ◽  
J. C. Lewis
Keyword(s):  
Whole Blood ◽  
Colloidal Gold ◽  
Blood Platelets ◽  
Cell Contact ◽  
Computer Assisted ◽  
Adherent Cells ◽  
Differential Centrifugation ◽  
Computer Reconstruction ◽  
Sds Page ◽  
Whole Mount

Thrombocytes (TC), the avian equivalent of blood platelets, support hemostasis by aggregating at sites of injury. Studies in our lab suggested that fibrinogen (fib) is a requisite cofactor for TC aggregation but operates by an undefined mechanism. To study the interaction of fib with TC and to identify fib receptors on cells, fib was purified from pigeon plasma, conjugated to colloidal gold and used both to facilitate aggregation and as a receptor probe. Described is the application of computer assisted reconstruction and stereo whole mount microscopy to visualize the 3-D organization of fib receptors at sites of cell contact in TC aggregates and on adherent cells.Pigeon TC were obtained from citrated whole blood by differential centrifugation, washed with Ca++ free Hank's balanced salts containing 0.3% EDTA (pH 6.5) and resuspended in Ca++ free Hank's. Pigeon fib was isolated by precipitation with PEG-1000 and the purity assessed by SDS-PAGE. Fib was conjugated to 25nm colloidal gold by vortexing and the conjugates used as the ligand to identify fib receptors.

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