Equicevian Points and Equiangular Lines of a Triangle in an Isotropic Plane

Kolar-Begović, Zdenka; Kolar-Šuper, Ružica; Volenec, Vladimir

doi:10.5644/sjm.11.1.08

Cited by 2 publications

(9 citation statements)

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“…the centroid is G = − 2 3 q, 0 , and the Euler line has the equation x = − 2 3 q, [6]. The properties of an allowable triangle in the isotropic plane have been studied in numerous papers, [3], [4], [5], [6], [7], [8], [14], [16], [18]. The observed triangle was settled in the standard position having the circumcircle with the equation y = x 2 and the vertices of the form A = (a, a 2 ), B = (b, b 2 ), C = (c, c 2 ), where a + b + c = 0.…”

Section: Covertex Inscribed Triangles Of Parabolamentioning

confidence: 99%

“…is the inscribed circle of the triangle ABC. The line BC has the equation x + ay + bc = 0 given in (4). The intersection point of BC and the circle (8) is the point D = −2bc, bc a having the intersection multiplicity 2.…”

Section: Covertex Inscribed Triangles Of Parabolamentioning

confidence: 99%

“…The concepts of equicevian points and equiangular lines of the triangle in the isotropic plane were studied in [4] where a number of their significant properties was considered. A point P such that the points D = AP ∩ BC, E = BP ∩ CA, F = CP ∩ AB satisfy the equalities…”

Section: Covertex Inscribed Triangles Of Parabolamentioning

confidence: 99%

“…The equicevian points of the standard triangle considered in [4] are so to say equal. It is not like that in the case of the covertex inscribed triangle of the parabola.…”

Section: Theorem 15mentioning

confidence: 99%

“…They are symmetrical with respect to the centroid G of the triangle and they lie on its Steiner axis. It is also known that they are the foci of its circumscribed Steiner ellipse of the triangle ABC, [4]. But in the case of the covertex inscribed triangle, it can be noticed that one of them, the point P 1 , lies on the vertex tangent (directrix) of parabola P .…”

Section: Theorem 15mentioning

confidence: 99%

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Covertex Inscribed Triangles of Parabola in Isotropic Plane

2019

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In the paper the concept of a covertex inscribed triangle of a parabola in an isotropic plane is introduced. It is a triangle inscribed to the parabola that has the centroid on the axis of parabola, i.e. whose circumcircle passes through the vertex of the parabola. We determine the coordinates of the triangle centers, and the equations of the lines, circles and conics related to the triangle.Covertex trokuti upisani paraboli u izotropnoj ravnini SAŽETAK U radu se uvodi pojam covertex trokuta upisanog paraboli u izotropnoj ravnini. To je trokut upisan paraboličije težište leži na osi parabole, tj.čija opisana kružnica prolazi tjemenom parabole. Odreduju se koordinate točaka te jednadžbe pravaca, kružnica i konika povezanih s tim trokutom.Ključne riječi: izotropna ravnina, trokut, parabola MotivationThe following theorem, which can be found in [10], is a well-known fact from the geometry of Euclidean plane:Let A, B,C be three points on a parabola P different from its vertex and different mutually. These are the equivalent statements:1 0 The normal lines to P at A, B,C are concurrent. 2 0 The centroid of the triangle ABC lies on the axis of parabola P .3 0 The circumcircle of the triangle ABC passes through the vertex of parabola P .The perpendicularity is not defined in the isotropic plane, and often an isotropic line plays a role of a line perpendicular to a given one. Therefore, every normal to P passes through the absolute point. From that point of view, the property 1 0 is fulfilled for any three points on the parabola, and it is interesting to study the triangles having properties 2 0 and 3 0 .The result above together with other results stated in [10] inspired the authors to write this paper.

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Section: Covertex Inscribed Triangles Of Parabolamentioning

confidence: 99%

Section: Covertex Inscribed Triangles Of Parabolamentioning

confidence: 99%

Section: Covertex Inscribed Triangles Of Parabolamentioning

confidence: 99%

“…The equicevian points of the standard triangle considered in [4] are so to say equal. It is not like that in the case of the covertex inscribed triangle of the parabola.…”

Section: Theorem 15mentioning

confidence: 99%

Section: Theorem 15mentioning

confidence: 99%

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Covertex Inscribed Triangles of Parabola in Isotropic Plane

2019

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Occurrence of ABO And RhD Incompatibility with Rh Negative Mothers

Izetbegović¹

2013

Mater Sociomed

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Introduction:Hemolytic disease of the newborn was first described in the medical literature 1609, when it was diagnosed in one French housewife. In 1932 Diamond and colleagues described the mutual relationship of fetal hydrops, jaundice, anemia and erythoblastosis, which was later called fetal erytroblastosis. Hemolytic disease of the newborn (HDN) in the strict sense is considered disease whose basis is accelerated immune destruction of fetal/child erythrocytes that are bound to IgG antibodies of maternal origin. These antibodies are directed against antigens of father’s origin, which are present in the fetal/children’s erythrocytes and that the mother’s immune system recognizes them as foreign antigens.Goal:The goal is that in the period from January 1st 2011 to October 23st 2013 determine the frequency of ABO and Rh D incompatibilities in our sample of pregnant women/mothers, and to underscore the importance of regular check of ABO Rh D negative pregnant women and application specific Rh D protection.Material and methods:In the General Hospital “Prim. Dr. Abdulah Nakas” in Sarajevo by retrospective study are followed several relevant variables. Immune alloantibodies were detected in vivo by indirect Coombs test (ICT) with serum mother and O test erythrocytes, by direct Coombs test (DCT) with erythrocytes of a newborn.Results:The total number of births ABO Rh D negative was 596 (14%) and ABO Rh D positive mothers 4261 (86%). Of the total number of Rh D negative mothers there was A Rh D: negative mothers 42%; O Rh D negative 33%; B Rh D: negative 17% and AB Rh D: negative 8%. Most of immune antibodies appear in mothers with O Rh D: negative blood type. The emergence of immune antibodies in the Rh D negative mothers was 1%, the appearance of ABO incompatibilities amounted to 2.3% of our sample.Conclusion:In order to reduce the occurrence of alloimmunization of the mother to erythrocyte antigens of the newborn that can lead to major complications in subsequent pregnancies of Rh D: negative mothers and HDN constant monitoring in order to prevent them is necessary. Prevention is essential because once immunized mother will remain immunized for life.

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Equicevian Points and Equiangular Lines of a Triangle in an Isotropic Plane

Cited by 2 publications

References 1 publication

Covertex Inscribed Triangles of Parabola in Isotropic Plane

Covertex Inscribed Triangles of Parabola in Isotropic Plane

Occurrence of ABO And RhD Incompatibility with Rh Negative Mothers

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