The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the

foundations of geometry Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but t ...

, introduced and studied by Friedrich Bachmann.. It states: Bachmann showed that, in the absence of the

Archimedean axiom In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typicall ...

, 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 In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...

, as shown by

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...

. In the presence of the

, the Lotschnittaxiom is equivalent with the

Equivalent formulations

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 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 that it is equivalent to the following axiom proposed by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaHenri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...

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: 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 and in.

Connection with the Parallel Postulate

As shown in, the conjunction of the Lotschnittaxiom and of Aristotle's axiom is equivalent to the

References

Sources

* * * *{{Citation, first1=Victor, last1=Pambuccian, first2=Celia, last2=Schacht, title= The ubiquitous axiom, journal=Results in Mathematics, volume=76, year=2021, issue=3, pages=1–39, doi=10.1007/s00025-021-01424-3, s2cid=236236967 , url=https://link.springer.com/article/10.1007/s00025-021-01424-3 Foundations of geometry