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Geometry, Volume 2, Issue 3 (September 2025) – 2 articles

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17 pages, 1278 KiB  
Review
The Multiple Utility of Kelvin’s Inversion
by Eleftherios Protopapas
Geometry 2025, 2(3), 11; https://doi.org/10.3390/geometry2030011 - 9 Jul 2025
Viewed by 78
Abstract
Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in R2, but it does not in Rn, for n>2. Lord Kelvin overcame this [...] Read more.
Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in R2, but it does not in Rn, for n>2. Lord Kelvin overcame this problem by defining a new (at the time) inversion, the so-called Kelvin’s inversion (or transformation). This inversion has many good properties, making it extremely useful in each case where the geometry of the original problem raises issues. But by using Kelvin’s inversion, these issues are transformed into easier ones, due to a simpler geometry. In this review paper, we study Kelvin’s inversion, deploying its basic properties. Moreover, we present some applications, where its use enables scientists to solve difficult problems in scattering, electrostaticity, thermoelasticity, potential theory and bioengineering. Full article
(This article belongs to the Special Issue Feature Papers in Geometry)
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9 pages, 609 KiB  
Article
On Yiu’s Equilateral Triangles Associated with a Kiepert Hyperbola
by Cherng-tiao Perng
Geometry 2025, 2(3), 10; https://doi.org/10.3390/geometry2030010 - 1 Jul 2025
Viewed by 154
Abstract
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are [...] Read more.
In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle. Full article
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