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Search Results (3)

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Keywords = isogonal points

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16 pages, 304 KB  
Article
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
Viewed by 1369
Abstract
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 ABC. 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 ABC. For example, the circumcenter is represented by the polynomial a(b2+c2−a2). It is not so well known that triangle centers have barycentric coordinates, such as tanA : 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(a2+b2−c2)(a2+c2−b2). 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)
10 pages, 2188 KB  
Article
Isoptic Point of the Non-Cyclic Quadrangle in the Isotropic Plane
by Ema Jurkin, Marija Šimić Horvath and Vladimir Volenec
Mathematics 2024, 12(22), 3610; https://doi.org/10.3390/math12223610 - 19 Nov 2024
Viewed by 695
Abstract
We study the non-cyclic quadrangle ABCD in the isotropic plane and its isoptic point. This is a continuation of the research carried out in a few previous papers. There, we put the non-cyclic quadrangle in the standard position, which enables [...] Read more.
We study the non-cyclic quadrangle ABCD in the isotropic plane and its isoptic point. This is a continuation of the research carried out in a few previous papers. There, we put the non-cyclic quadrangle in the standard position, which enables us to prove its properties using a simple analytical method. In the standard position, the special hyperbola xy=1 is circumscribed to the quadrangle. Hereby, we use the same method to obtain several results related to the isoptic point of the non-cyclic quadrangle. The isoptic point T is the inverse image of points A′, B′, C′, D′ with respect to circumcircles of BCD, ACD, ABD, ACD, respectively, where A′, B′, C′, D′ are isogonal points to vertices A, B, C, D with respect to triangles BCD, ACD, ABD, ACD. The circumircles are seen from T under the equal angles. Our analysis is motivated by the Euclidean results already published in the literature. Full article
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17 pages, 377 KB  
Article
Isoptic Point of the Complete Quadrangle
by Ema Jurkin, Marija Šimić Horvath and Vladimir Volenec
Axioms 2024, 13(6), 349; https://doi.org/10.3390/axioms13060349 - 24 May 2024
Cited by 1 | Viewed by 983
Abstract
In this paper, we study the complete quadrangle. We started this investigation in a few of our previous papers. In those papers and here, the rectangular coordinates are used to enable us to prove the properties of the rich geometry of a quadrangle [...] Read more.
In this paper, we study the complete quadrangle. We started this investigation in a few of our previous papers. In those papers and here, the rectangular coordinates are used to enable us to prove the properties of the rich geometry of a quadrangle using the same method. Now, we are focused on the isoptic point of the complete quadrangle ABCD, which is the inverse point to A′,B′,C′, and D′ with respect to circumscribed circles of the triangles BCD, ACD, ABD, and ABC, respectively, where A′,B′,C′, and D′ are isogonal points to A,B,C, and D with respect to these triangles. In studying the properties of the quadrangle regarding its isoptic point, some new results are obtained as well. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
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