Geometry, Volume 1, Issue 1 (December 2024) – 4 articles

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4 pages, 404 KiB  
Article
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 255
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  
Article
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 865
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 [...] Read more.
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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13 pages, 303 KiB  
Article
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
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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 u1(X)=(yz)/x:(zx)/y:(xy)/z. For each ui, there exist, formally, two points, P and U, such that ui(P)=ui(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, ui(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  
Editorial
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 770
Abstract
In the ancient realm of geometry, we have witnessed the ultimate display of mathematical abstract thought [...] Full article
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