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On relatively short and long sides of convex pentagons 总被引:1,自引:0,他引:1
By the relative distance of pointsa andb of a convex bodyC we mean the ratio of the Euclidean distance ofa andb to the half of the Euclidean distance ofa, b C such thatab is a longest chord ofC parallel to the segmentab. We say that a sideab of a convexn-gon is relatively short (respectively: relatively long) if the relative distance ofa andb is at most (respectively: at least) the relative distance of two consecutive vertices of the regularn-gon. We show that every convexn-gon, wheren 5, has a relatively short side and a relatively long side, and that it is affine-regular if and only if all its sides are of equal relative lengths.Research supported in part by Komitet Bada Naukowych (Committee of Scientific Research), grant number 2 2005 92 03. 相似文献
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Marshall Lassak 《International Journal of Mathematical Education in Science & Technology》2013,44(5):690-696
Using dynamic graphs, future secondary mathematics teachers were able to represent and communicate their understanding of a brief mathematical investigation in a way that a symbolic proof of the problem could not. Four different student work samples are discussed. 相似文献
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We prove that the unit disk C of an arbitrary Minkowski plane contains an equilateral triangle in at least one of the orientations, whose oriented side lengths are ${\frac{3}{2}}$ . We also prove that C permits to inscribe a triangle whose sides are of lengths at least ${\frac{3}{2}}$ in the positive orientation, or that they are of lengths at least ${\frac{3}{2}}$ in the negative orientation. The ratio ${\frac{3}{2}}$ in both the theorems is best possible. 相似文献
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The paper deals with the problem of approximating point sets byn-point subsets with respect to the minimal widthw. Let, in particular,
d
denote the family of all convex bodies in Euclideand-space, letA
d
and letn be an integer greater thand. Then we ask for the greatest number =
n
(A) such that everyA A contains a polytope withn vertices which has minimal width at least w(A). We give bounds for
n
(
d
), for
n
(2133;
d
), and for
n
(W
d
), where 2133;
d
,W
d
denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.Dedicated to Professor L. danzer on the occasion of his sixtieth birthdayResearch for this paper was conducted in the academic year 1986/87 while both authors were visiting the University of Washington, Seattle. P. Gritzmann was supported by the Alexander von Humboldt Foundation. 相似文献
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Brophy JJ Goldsack RJ Palá-Paúl J Copeland LM Lassak EV 《Natural product communications》2010,5(11):1833-1836
The essential oils from the aerial parts of three rare Australian endemic species of Darwinia have been extracted by hydrodistillation and analyzed by gas chromatography (GC) and gas chromatography coupled to mass spectrometry (GC-MS). In D. procera, myrtenyl acetate (6.1-29.6%), alpha-pinene (6.9-25.1%), gamma-terpinene (6.2-13.6%), bicyclogermacrene (5.5-10.8%) and (E)-nerolidol (3.4-9.7%) were the principal components detected. D. fascicularis ssp. fascicularis produced an oil in which (E)-nerolidol (33.0%), alpha-pinene (15.1%) and gamma-terpinene (10.2%) were the principal components. In D. peduncularis the major constituents were alpha-pinene (33.5%), gamma-terpinene (23.1%) and bicyclogermacrene (6.7%). 相似文献
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