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• W2289328893 abstract "Starting with an analysis of the configuration of chords of c ontact points with two lines, defined on conics circumscribing a tri angle and tangent to these lines, we prove properties relating to the case the con ics are parabolas and a resulting method to construct the parabola tangent to four lines. It is well known ((3, p. 42), (10, p. 184), (7, II, p. 256)), tha t given three points A, B, C and two lines in general position, there are either none or fo ur conics passing through the points and tangent to the given lines. A light simplification of Chasles notation ((2, p. 304)) for these curves is 3p2t conics. The conics exist if either the two lines do not intersect the interior of the tr iangle ABC or the two lines intersect the interior of the same two sides of ABC. In all other cases there are no conics satisfying the above requirements. In this art icle, we obtain a formal condition (Theorem 6) for the existence of these conics, relating to the geometry of the triangle ABC. In addition we study the configuration of a triangle and two l ines satisfying certain conditions. In §2 we introduce the middle-tripolar, which plays a key role in the study. In §3 we review the properties of generalized quadratic transforms, which are relevant for our discussion. In §§4, 5 we relate the classical result of existence of 3p2t conics to the geometry of the triangle ABC. In the two last sections we prove related properties and construction methods for parabolas. 2. The middle-tripolar If a parabola circumscribes a triangle ABC and is tangent to a line l (at a point different from the vertices), then l does not intersect the interior of ABC. In this section we obtain a characterization of such lines. For this , we start with a point P on the plane of triangle ABC and consider its traces A1, B1, C1 and their har- monic conjugates A2, B2, C2, with respect to the sides BC, CA, AB, later lying on the tripolar tr(P ) of P (See Figure 1). By applying Newton's theorem ((5, p. 62)) on the diagonals of the quadrilateral A1B1B2A2 we see that the middles A ' , B ' , Crespectively of the segments A1A2, B1B2, C1C2 are on a line, which I call the middle-tripolar of the point P and denote by mP . In the following dis- cussion a crucial role plays a certain symmetry among the four lines defined by the sides of the cevian A1B1C1 of P and the tripolar tr(P ), in relation to the har- monic associates ((13, p. 100)) P1, P2, P3 of P. It is, namely, readily seen that for each of these four points the corresponding sides of cevian triangle and tripolar define the same set of four lines. A consequence of this fact is that all four points P, P1, P2, P3 define the same middle-tripolar, which lies totally in the ex terior of" @default.
• W2289328893 created "2016-06-24" @default.
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• W2289328893 date "2012-01-01" @default.
• W2289328893 modified "2023-09-27" @default.
• W2289328893 title "On Tripolars and Parabolas" @default.
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