Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of ...triangles is called a poristic system of triangles. We study the trace of the centroid, the Feuerbach point, the symmedian point, the Gergonne point, the Steiner point and the Brocard points for such a system of triangles. We also study the traces of some further points associated with the triangles of the poristic family, and we prove that the vertices of the contact triangle, tangential triangle and anticomplementary triangle move on circles while the initial triangle traverses the poristic family.
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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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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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.
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 present educational material which is a basis for a web textbook designed for the lectures on geometric subjects. The textbook is available online for free. It differs from previous ...textbooks in the application of information technologies in presenting classic contents and introducing 3D modelling in descriptive geometry teaching.
In this paper we define q-spherical surfaces as the surfaces that contain the
absolute conic of the Euclidean space as a q?fold curve. Particular
attention is paid to the surfaces with singular ...points of the highest order.
Two classes of such surfaces, with one and two n?fold points, are discussed
in detail. We study their properties, give their algebraic equations and
visualize them with the program Mathematica.
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.
In this paper we define and construct a new class of algebraic surfaces in
three-dimensional Euclidean space generated by a curve and a congruence of
circles. We study their properties and visualize ...them. For computing and
plotting, we use the program Wolfram Mathematica.
nema
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