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Definition: Two distinct *points* \(x\) and \(z\) in a Euclidean space (simply called a space in the following) viewed as a subspace of \(\mathbb{R}^{n}\) with \(n \in {}^{\omega }\mathbb{N}^{*}\) (see Set Theory) are said to be a *pair (of points)*. A *line segment* is a pair \((x, z)\) together with all *inner points* \(y\) in the space that are distinct from both the *starting point* \(x\) and the *end point* \(z\) and that lie *between* \(x\) and \(z\), satisfying \(||x - y|| + ||y - z|| = ||x - z||\) with respect to the Euclidean norm \(||\cdot||\).

Definition: Two line segments are said to *intersect* if they have precisely one point in common. This includes the case when the common point is only found after completing each line segment with all other inner points of that segment within \(\mathbb{R}^{n}\). A (one-dimensional) set of points in the space with the property that each point has at least one and at most two gaplessly neighbouring points is called a *line*, and a maximal two-dimensional subspace is called a *plane*.

Definition: A line segment is said to be a *straight line* if both its starting point and its end point lie on the boundary of the space, for the time being with the additional requirement that none of its inner points do. Two line segments are said to be *parallel* if one line may be obtained from the other by means of a translation or the minimum distances between each point on one line segment and the other line segment are identical. Any line segment in the space parallel to one of the straight lines defined above is also called a straight line.

Result: By defining short straight lines, arbitrarily many counterexamples can be given based on the above to Pasch's axiom, the axiom of line completeness, as well as various other axioms and their equivalents. The parallel postulate is redundant in Euclidean Geometry, if the parallel straight line running through a given point is uniquely defined by the shortest distance from the original straight line.

If two straight lines are only considered to be parallel when they lie in the same plane and do not intersect, then the parallel postulate does not hold when the reciprocal of the distance between the straight line and the given point is greater than infinity or smaller than \(|{}^{\omega }\mathbb{N}|\), since then infinitely many distinct straight lines can be found that pass through the given point without intersecting the original straight line.

The Archimedean axiom must be extended to the case where a segment is marked off an infinite natural number of times without exceeding the starting point or end point of a straight line, or replaced by the Archimedean theorem (in the finite case). Pasch's axiom is also unnecessary, since every straight line must be fully contained in the interior of some triangle due to its maximum length, and hence so must its boundary.

Toeplitz' conjecture: Every Jordan curve admits an inscribed square.

Counterexamples: The right-angled triangle with two sides of length \(d0\) and the obtuse triangle where we infinitesimally move a vertex of at most one inscribed square within the limits.\(\square\)

Theorem: There is a Jordan domain with more than one equichordal point (cf. [931], p. 9 f.).

Proof: We infinitesimally juxtapose equichordal points within the limits.\(\square\)

Fickett's theorem: For any relative positions of two overlapping congruent rectangular \(n\)-prisms \(Q\) and \(R\) with \(n \in {}^{\omega }\mathbb{N}_{\ge 2}\), we have for the exact standard measure \(\mu\) (see Topology and [931], p. 25), where \(\mu\) for \(n = 2\) needs to be replaced by the Euclidean path length \(L\), that:\[1/(2n - 1) < r := \mu(\partial Q \cap R)/\mu(\partial R \cap Q) < 2n - 1.\]Proof: Since the underlying extremal problem has its maximum for rectangles with the side lengths \(s\) and \(s + 2d0\), min \(r = s/(3s - 2d0) \le r \le\) max \(r = (3s - 2d0)/s\) holds. The proof for \(n > 2\) is analogous.\(\square\)

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