In this paper we define and study Morley’s triangles of a triangle in the isotropic plane. We derive the equations of angle trisectors of angles of the standard triangle in the isotropic plane, and ...utilizing them we get the coordinates of vertices of Morley’s triangles of the standard triangle. We investigate relationships between Morley’s triangles and the initial triangle, as well as connections with some other triangle elements in the isotropic plane. Finally, we consider some dual concepts of the introduced concepts.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
2.
On quadruples of orthopoles Volenec, Vladimir; Jurkin, Ema; Horvath, Marija Šimić
Journal of geometry,
12/2023, Volume:
114, Issue:
3
Journal Article
Peer reviewed
In this paper we study a complete quadrangle in the Euclidean plane that has a rectangular hyperbola circumscribed to it. Hereby, the approach is based on the rectangular coordinates and we prove the ...following main result: Let
ABCD
be a complete quadrangle and
l
a
,
l
b
,
l
c
,
l
d
mutually parallel lines through the circumcenters of
BCD
,
ACD
,
ABD
,
ABC
, respectively. Orthopoles of the lines
l
a
,
l
b
,
l
c
,
l
d
with respect to the triangles
BCD
,
ACD
,
ABD
,
ABC
lie on a line which passes through the center of the rectangular hyperbola
H
circumscribed to
ABCD
, and it is antiparallel to the given lines with respect to the axes of the hyperbola
H
.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of ...tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank 2, we derive the Hessian configuration (123,164) of points and lines. Finally, we introduce and investigate the de Vries configuration of points and lines in a cubic structure.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
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.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we study the properties of a complete quadrangle in the Euclidean plane. The proofs are based on using rectangular coordinates symmetrically on four vertices and four parameters ...a,b,c,d. Here, many properties of the complete quadrangle known from earlier research are proved using the same method, and some new results are given.
In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the ...first Brocard triangle is homologous to the given triangle and that these two triangles are parallelogic. We consider the relationships between the first Brocard triangle and the Steiner axis, the Steiner point, and the Kiepert parabola of the triangle. We also investigate some other interesting properties of this triangle and consider relationships between the Euclidean and the isotropic case.
In this paper, we introduce and study new geometric concepts in a general cubic structure. We define the concept of the inflection point in a general cubic structure and investigate relationships ...between inflection points and associated and corresponding points in a general cubic structure.
In this paper, we consider the Feuerbach point and the Feuerbach line of a triangle in the isotropic plane, and investigate some properties of these concepts and their relationships with other ...elements of a triangle in the isotropic plane. We also compare these relationships in Euclidean and isotropic cases.
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