Dieter Ruoff, Ascending lines in the hyperbolic plane,
Forum Geometricorum, 16 (2016) 125--132.

Abstract. On the basis of the familiar proportionality theorems a line in the Euclidean plane, which ascends from a horizontal base, can be assigned a constant slope. In a non-Euclidean setting (where the proportionality theorems do not hold) this is not possible: A line segment begins its ascent more slowly than it finishes it, failing to reach at its midpoint half its final height. After reviewing two proofs of this fact we expand on it by comparing the ascent of different line segments. It is hoped that the results presented here, which belong to elementary synthetic non-Euclidean geometry, will contribute to enriching the offerings in the pertinent textbooks.

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