Geometry

16 pages, 304 KiB

Open AccessArticle

Trigonometric Polynomial Points in the Plane of a Triangle

by Clark Kimberling and Peter J. C. Moses

Geometry 2024, 1(1), 27-42; https://doi.org/10.3390/geometry1010005 - 23 Dec 2024

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It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths

a, b,

and c of a triangle

A B C

. For example, the circumcenter is represented by [...] Read more.

It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths

a, b,

and c of a triangle

A B C

. For example, the circumcenter is represented by the polynomial

a (b^{2} + c^{2} - a^{2})

. It is not so well known that triangle centers have barycentric coordinates, such as

\tan A : \tan B : \tan C

, that are also representable by polynomials, in this case, by

p (a, b, c) : p (b, c, a) : p (c, a, b)

, where

p (a, b, c) = a (a^{2} + b^{2} - c^{2}) (a^{2} + c^{2} - b^{2})

. This paper presents and discusses the polynomial representations of triangle centers that have barycentric coordinates of the form

f (a, b, c) : f (b, c, a) : f (c, a, b)

, where f depends on one or more of the functions in the set

{\cos, \sin, \tan, \sec, \csc, \cot}

. The topics discussed include infinite trigonometric orthopoints, the n-Euler line, and symbolic substitution. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

4 pages, 404 KiB

Open AccessArticle

Hagge Configurations and a Projective Generalization of Inversion

by Zoltán Szilasi

Geometry 2024, 1(1), 23-26; https://doi.org/10.3390/geometry1010004 - 12 Nov 2024

Viewed by 497

Abstract

In this article, we provide elementary proofs of two projective generalizations of Hagge’s theorems. We describe Steiner’s correspondence as a projective generalization of inversion. Full article

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7 pages, 3051 KiB

Open AccessArticle

Packing Series of Lenses in a Circle: An Area Converging to 2/3 of the Disc

by Andrej Hasilik

Geometry 2024, 1(1), 16-22; https://doi.org/10.3390/geometry1010003 - 5 Aug 2024

Viewed by 1069

Abstract

We describe a series of parallel lenses with constant proportions packed in a circle. To construct n lenses, a regular 2(n + 1)-gon is drawn with a central diagonal of 2r length, followed by an array of n parallel diagonals perpendicular to the former. These diagonals and the central angle of the pair of peripherals, the shortest diagonals, are used to construct n rhombi. The rhombi define the shape of lenses tangential to them. To construct the arcs of the lenses, beams perpendicular to the sides of each rhombus are drawn. Four beams radiating from the top and bottom vertices of each rhombus intersect in the centers of a pair of coaxal circles. Thus, the vertical axis of each rhombus coincides with the radical axis of the pair. The intersection of the pair represents the corresponding lens. All n lenses form a tangential sequence along the central diagonal. Their cusps circumscribe the polygon and the lenses themselves. The area covered by the lenses converges to (2/3) πr². Full article

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Graphical abstract

13 pages, 303 KiB

Open AccessArticle

Unary Operations on Homogeneous Coordinates in the Plane of a Triangle

by Peter J. C. Moses and Clark Kimberling

Geometry 2024, 1(1), 3-15; https://doi.org/10.3390/geometry1010002 - 8 Jul 2024

Viewed by 1030

Abstract

Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear)

x : y : z

. Eight unary operations discussed in this paper include [...] Read more.

Suppose that X is a triangle center with homogeneous coordinates (barycentric or trilinear)

x : y : z

. Eight unary operations discussed in this paper include

u_{1} (X) = (y - z) / x : (z - x) / y : (x - y) / z

. For each

u_{i}

, there exist, formally, two points, P and U, such that

u_{i} (P) = u_{i} (U) = X

. To such pairs of inverses are applied nine binary operations, each resulting in a triangle center. If L is a line, then formally,

u_{i} (L)

is a cubic curve that passes through the vertices

A, B, C

. If L passes through the point

1 : 1 : 1

(the centroid or incenter, assuming that the coordinates are barycentric or trilinear), then the cubic is degenerate as the union of a parabola and the line at infinity. The methods in this work are largely algebraic and computer-dependent. Full article

2 pages, 297 KiB

Open AccessEditorial

Geometry: A Bridge Connecting All Things

by Yang-Hui He

Geometry 2024, 1(1), 1-2; https://doi.org/10.3390/geometry1010001 - 29 May 2024

Viewed by 1066

Abstract

In the ancient realm of geometry, we have witnessed the ultimate display of mathematical abstract thought [...] Full article

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Geometry, Volume 1, Issue 1 (December 2024) – 5 articles

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