Abstract
The following characterizations of the Euclidean plane are obtained: the two tangent segments of the unit circle of a normed plane from each point of a disc centered at the origin with sufficiently large diameter have equal lengths; the lengths of the tangent segments from each point of a fixed circle centered at the origin are determined only by the radius of this circle. Three further characterizations of the Euclidean plane are obtained by considering properties of certain points related to an exterior point of the unit disc and the two tangent segments corresponding to it. To obtain one of these characterizations, the notion of arc-length orthogonality is introduced, and the Euclidean plane is also characterized via a relation between arc-length orthogonality and Birkhoff orthogonality.
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The work of S. Wu is supported by the National Natural Science Foundation of China (grant number 11001068), a foundation from Ministry of Education of Heilongjiang Province for Returned Overseas Chinese Scholars (grant number 1251H013), the Scientific Research Foundation for the Returned Overseas Chinese Scholars from State Education Ministry, a foundation from Harbin University of Science and Technology (grant number 2009YF028), and by Deutsche Forschungsgemeinschaft.
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Martini, H., Wu, S. Tangent segments and orthogonality types in normed planes. J. Geom. 99, 89–100 (2010). https://doi.org/10.1007/s00022-011-0063-5
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DOI: https://doi.org/10.1007/s00022-011-0063-5