Lotschnittaxiom
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The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann.[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.
It was shown in[3] that it is also equivalent to the statement
The altitude in an isosceles triangle with base angles of 45° is less than the base.
and in [4] that it is equivalent to the following axiom proposed by Lagrange:[5]
If the lines a and b are two intersecting lines that are parallel to a line g, then the reflection of a in b is also parallel to g.
As shown in,[6] the Lotschnittaxiom is also equivalent to the following statements, the first one due to A. Lippman, the second one due to Henri Lebesgue [7]
Given any circle, there exists a triangle containing that circle in its interior.
Given any convex quadrilateral, there exists a triangle containing that convex quadrilateral in its interior.
Three more equivalent formulations, all purely incidence-geometric, were proved in:[8]
Given three parallel lines, there is a line that intersects all three of them.
There exist lines a and b, such that any line intersects a or b.
If the lines a_1, a_2, and a_3 are pairwise parallel, then there is a permutation (i,j,k) of (1,2,3) such that any line g which intersects a_i and a_j also intersects a_k.
In Bachmann's geometry of line-reflections
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]