In this paper we focus on the formalization of the proofs of equivalence between different versions of Euclid's 5 th postulate. Our study is performed in the context of Tarski's neutral geometry, or equivalently in Hilbert's geometry defined by the first three groups of axioms, and uses an intuitionistic logic, assuming excluded-middle only for point equality. Our formalization provides a clarification of the conditions under which different versions of the postulates are equivalent. Following Beeson, we study which versions of the postulate are equivalent, constructively or not. We distinguish four groups of parallel postulates. In each group, the proof of their equivalence is mechanized using intuitionistic logic without continuity assumptions. For the equivalence between the groups additional assumptions are required. The equivalence between the 34 postulates is formalized in Archimedean planar neutral geometry. We also formalize a theorem due to Szmiliew. This theorem states that, assuming Archimedes' axiom, any statement which hold in the Euclidean plane and does not hold in the Hyperbolic plane is equivalent to Euclid's 5 th postulate. To obtain all these results, we have developed a large library in planar neutral geometry, including the formalization of the concept of sum of angles and the proof of the Saccheri-Legendre theorem, which states that assuming Archimedes' axiom, the sum of the angles in a triangle is at most two right angles.

This paper describes the formalization of the arithmetization of Euclidean plane geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of the book by Schwabhäuser Szmielew and Tarski Metamathematische Methoden in der Geometrie. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method. Moreover, we solve a challenge proposed by Beeson: we prove that, given two points, an equilateral triangle based on these two points can be constructed in Euclidean Hilbert planes. Finally, we derive the axioms for another automated deduction method: the area method.

This paper describes the formalization of the arithmetization of Euclidean geometry in the Coq proof assistant. As a basis for this work, Tarski's system of geometry was chosen for its well-known metamathematical properties. This work completes our formalization of the two-dimensional results contained in part one of [SST83]. We defined the arithmetic operations geometrically and proved that they verify the properties of an ordered field. Then, we introduced Cartesian coordinates, and provided characterizations of the main geometric predicates. In order to prove the characterization of the segment congruence relation, we provided a synthetic formal proof of two crucial theorems in geometry, namely the intercept and Pythagoras' theorems. To obtain the characterizations of the geometric predicates, we adopted an original approach based on bootstrapping: we used an algebraic prover to obtain new characterizations of the predicates based on already proven ones. The arithmetization of geometry paves the way for the use of algebraic automated deduction methods in synthetic geometry. Indeed, without a "back-translation" from algebra to geometry, algebraic methods only prove theorems about polynomials and not geometric statements. However, thanks to the arithmetization of geometry, the proven statements correspond to theorems of any model of Tarski's Euclidean geometry axioms. To illustrate the concrete use of this formalization, we derived from Tarski's system of geometry a formal proof of the nine-point circle theorem using the Gröbner basis method.

Abstract. We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof. §1. Introduction. We intend this paper to be read by mathematicians who are unfamiliar with mathematical logic and also unfamiliar with non-Euclidean geometry; therefore we ask the patience of readers who are familiar with one or both of these subjects.We begin with a brief discussion of axioms for plane Euclidean geometry. Every such axiom system will have variables for points. Some axiom systems may have variables for other objects, such as lines or angles, but Tarski showed that these are not really necessary. For example, angles can be discussed in terms of ordered triples of points, and lines in terms of ordered pairs of points. For simplicity we focus on such a points-only axiomatization.The primitive relations of such a theory usually include a "betweenness" relation, and an "equidistance" relation. We write T(a, b, c) to express that b lies (non-strictly) between a and c (on the same line), and E(a, b, c, d) to express that segment ab is congruent to segment cd. E stands for "equidistance", because in the standard model "congruent" means that the distance ab is equal to the distance cd; but there is nothing in the axioms about numbers to measure distance, or about distance itself. Sometimes it is convenient to use B(a, b, c) for strict betweenness, i.e. a = b and b = c and T (a, b, c).Some of the axioms will assert the existence of "new" points that are constructed from other "given" points in various ways. For example, one axiom says that segment ab can be extended past b to a point x, lying on the line determined by ab, such that segment bx is congruent to a given segment pq. That axiom can be written formally, using the logician's symbol ∧ for "and", asIt is possible to replace the quantifier ∃ with a "function symbol". We denote the point x that is asserted to exist by ext (a, b, p, q). Then the axiom looks like T(a, b, ext(a, b, p, q)) ∧ E(b, ext(a, b, p, q), p, q)

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