The notion of the Gergonne point of a triangle in the Euclidean plane is very well known, and the study of them in the isotropic setting has already appeared earlier. In this paper, we give two ...generalizations of the Gergonne point of a triangle in the isotropic plane, and we study several curves related to them. The first generalization is based on the fact that for the triangle ABC and its contact triangle AiBiCi, there is a pencil of circles such that each circle km from the pencil the lines AAm, BBm, CCm is concurrent at a point Gm, where Am, Bm, Cm are points on km parallel to Ai,Bi,Ci, respectively. To introduce the second generalization of the Gergonne point, we prove that for the triangle ABC, point I and three lines q1,q2,q3 through I there are two points G1,2 such that for the points Q1,Q2,Q3 on q1,q2,q3 with d(I,Q1)=d(I,Q2)=d(I,Q3), the lines AQ1,BQ2 and CQ3 are concurrent at G1,2. We achieve these results by using the standardization of the triangle in the isotropic plane and simple analytical method.
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.
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.
On quadruples of orthopoles Volenec, Vladimir; Jurkin, Ema; Horvath, Marija Šimić
Journal of geometry,
12/2023, Letnik:
114, Številka:
3
Journal Article
Recenzirano
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
.
The authors have studied the curvature of the focal conic in the isotropic plane and the form of the circle of curvature at its points has been obtained. Hereby, we discuss several properties of such ...circles of curvature at the points of a parabola in the isotropic plane.
U radu dajemo pregled nekih svojstava potpunog četverovrha ABCD u euklidskoj ravnini. Proučavamo kružnice s promjerima AB, AC, AD, BC, BD, CD, kao i nožišne trokute i nožišne kružnice točaka A, B, C, ...D s obzirom na trokute BCD, ACD, ABD, ABC redom navedene. Svi prikazani rezultati su poznati iz literature, ali ih ovdje dokazujemo koristeći istu metodu.
This paper presents an overview of some properties of a complete quadrangle ABCD in the Euclidean plane. We study the circles with diameters AB, AC, AD, BC, BD, and CD, as well as the pedal triangles and the pedal circles of the points A, B, C, D with respect to the triangles BCD, ACD, ABD and ABC, respectively. The presented results are known in literature, but here we prove them using a single method.
Geometry of the non cyclic quadrangle in the isotropic plane was introduced in 2 and 6. Herein, its diagonal triangle is studied and some nice properties of it are given.
Geometrija netetivnog ...četverovrha u izotropnoj ravnini uvedena je u člancima 2 and 6. Ovdje se proučava dijagonalni trokut i daju se neka njegova lijepa svojstva.
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