[1] It states: Perpendiculars raised on each side of a right angle intersect.Bachmann showed that, in the absence of the Archimedean axiom, it is strictly weaker than the rectangle axiom, which states that there is a rectangle, which in turn is strictly weaker than the Parallel Postulate, as shown by Max Dehn.
[2] In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate.
As shown by Bachmann, the Lotschnittaxiom is equivalent to the statement Through any point inside a right angle there passes a line that intersects both sides of the angle.
Its role in Friedrich Bachmann's absolute geometry based on line-reflections, in the absence of order or free mobility (the theory of metric planes) was studied in [9] and in.
[10] As shown in,[3] the conjunction of the Lotschnittaxiom and of Aristotle's axiom is equivalent to the Parallel Postulate.