Butterfly spirals Gabour, Manal
International journal of mathematical education in science and technology,
11/2022, Letnik:
53, Številka:
12
Journal Article
Recenzirano
In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite ...instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those sequences reveals bounded piecewise spirals, and is done by using the free dynamic mathematics software GeoGebra. It is also supported with rigorous reasoning concerning the convergence of a special sequence. Moreover, it raises related problems that encourage further investigation. This article suggests a content that can be useful for students at secondary as well as undergraduate levels since it incorporates principles taught to students during their first courses in Algebra, Calculus and Geometry especially those related to convergent sequences, the algebra of complex numbers, plane isometries and basic Euclidean geometry.
We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the last year of his life. Furthermore we add a ...projective Butterfly theorem which covers all known affine cases.
The butterfly theorem is proved by assigning point masses to the four vertices of the wings and using the distributive property of the mass centre of a mechanical system.
A real affine plane A_2 is called an isotropic plane I_2, if in A_2 a metric is induced by an absolute {f, F}, consisting of the line at infinity f of A_2 and a point $F\in f$.
Better butterfly ...theorem is one of the generalisations of the well-known butterfly theorem (1, 4). In this paper the better butterfly theorem has been adapted for the isotropic plane and its validity in I_2 has been proved.
Up till now the validity of the Butterfly theorem has been verified in
the Euclidean, isotropic, hyperbolic and pseudo-Euclidean plane. Furthermore, it
has been shown that an infinite number of ...butterfly points, located on a conic, is
associated with any quadrangle inscribed into a circle.
In the present paper we study the curve formed by butterfly lines. In the Euclidean
plane this curve is always a curve of order four and class three having one
real cusp while in the pseudo-Euclidean plane it can also be a curve of order four and
class three having three real cusps or a special parabola.
Leptiri u izotropnoj ravnini Beban Brkić, Jelena; Volenec, Vladimir
KoG,
02/2005, Letnik:
8., Številka:
8.
Paper
Odprti dostop
Realna afina ravnina A2 se naziva izotropnom ravninom I2 ako je metrika u A2 inducirana apsolutnom figurom {f, F}, koja se sastoji od neizmjerno dalekog pravca f ravnine A2 i točke F∈ f. U ovom je ...radu poznati Leptirov teorem smješten u izotropnu ravninu. Za taj teorem, kojeg od sada nazivamo Izotropnim leptirovim teoremom, dana su četiri dokaza.
Reversion Porisms in Conics HALBEISEN, Lorenz; HUNGERBÜHLER, Norbert; SCHİLTKNECHT, Marco
International Electronic Journal of Geometry,
10/2021, Letnik:
14, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We give a projective proof of the butterfly porism for cyclic quadrilaterals and present a general reversion porism for polygons with an arbitrary number of vertices on a conic. We also investigate ...projective properties of the porisms.
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