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• W2194906933 abstract "In this paper non-tangential quadrilaterals in the plane are studied. A quadrilateral is called standard if a parabola with the equation x = y 2 is inscribed in it. The properties of the standard quadrilateral related to the focus and the median of the quadrilateral are presented. Furthermore, the orthic of the quadrilateral is introduced. AMS subject classiflcations: 51N25 A projective plane with the absolute (in the sense of Cayley-Klein) consisting of a line ! and a point › on ! is called an plane. Usually such an a-ne coordinate system is chosen that in homogeneous coordinates ! : z = 0, i.e., ! is the line at inflnity, and › = (0;1;0). The line ! is said to be the absolute line and the point › the absolute point. A line incident with the absolute point is called an line, and a conic touching ! at the absolute point › is an circle (sometimes isotropic will be omitted). All the notions related to the geometry of the plane can be found in (7) and (8). Let A;B;C;D be any four lines in the plane where any two of them are not parallel and any three of them do not pass through the same point. The flgure consisting of these four lines and their six points of intersection is called a complete quadrilateral and will be denoted by ABCD. The lines A;B;C;D are the sides of the quadrilateral, and AB;AC;AD, B C;B D;C D are its vertices. Pairs of vertices A B;C D; A C;B D and A D;B C are called opposite. There is exactly one conic touching the lines A;B;C;D and the absolute line !. If this conic touches ! at the point ›, then it is an circle inscribed in ABCD. Otherwise, if the contact point ' is difierent from ›, then the conic is a parabola inscribed in ABCD. In this case the quadrilateral ABCD is non-tangential. Let us denote this parabola by P. Except for the tangent !, the parabola P has another tangent T , from the point ›, being an line. This line is called the directrix of the parabola P and its point T of contact with the parabola is called the focus of the parabola P. The joint line of the focus T and the point at inflnity ' is called the" @default.
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• W2194906933 date "2010-06-10" @default.
• W2194906933 modified "2023-09-23" @default.
• W2194906933 title "The focus and the median of a non-tangential quadrilateral in the isotropic plane" @default.
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