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This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given that there are no algebraic structures that one could associate with those minimal axiom systems.

This is a survey of axiom systems for fragments of naturally encountered geometries which are just barely strong enough to imply that there are infinitely many objects in the universe of any of its models.

We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann’s Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

We prove that both Lippmann’s axiom of 1906, stating that for any circle there exists a triangle circumscribing it, and Lebesgue’s axiom of 1936, stating that for every quadrilateral there exists a triangle containing it, are equivalent, with respect to Hilbert’s plane absolute geometry, to Bachmann’s Lotschnittaxiom, which states that perpendiculars raised on the two legs of a right angle meet. We also show that, in the presence of the Circle Axiom, the statement “There is an angle such that the perpendiculars raised on its legs at equal distances from the vertex meet” is equivalent to the negation of Hilbert’s hyperbolic parallel postulate.

This is a survey of results from the rich theory of medial and more general quasigroups and its close connection with the geometry of point-reflections. The emphasis is on questions regarding the simplest axiomatization, in the sense of the minimal number of variables appearing in the identtties, on dependence or independence of axioms, and on representation theorems in the style of Toyoda.

Absolute planes in which the elementary Archimedean axiom holds satisfy Aristotle’s axiom. Absolute planes satisfying both the elementary Archimedean axiom and Bachmann’s Lotschnittaxiom are Euclidean. The Corollary to Aristotle’s axiom is equivalent to Aristotle’s axiom.

Solving a problem left open in Hajja and Martini (Mitt. Math. Ges. Hamburg 33:135–159, 2013), we prove, inside a weak plane absolute geometry, that, for every point P in the plane of a triangle ABC there exists a point Q inside or on the sides of ABC which satisfies: 1 $$\begin{aligned} AQ \le AP, BQ \le BP, CQ\le CP. \end{aligned}$$ If P lies outside of the triangle ABC, then Q can be chosen to both lie inside the triangle ABC and such that the inequalities in (1) are strict. We will also provide an algorithm to construct such a point Q.

By looking at concrete examples from elementary geometry, we analyse the manner in which the simplicity of proofs could be defined. We first find that, when presented with two proofs coming from mutually incompatible sets of assumptions, the decision regarding which one is simplest can be made, if at all, only on the basis of reasoning outside of the formal aspects of the axiom systems involved. We then show that, if the axiom system is fixed, a measure of proof simplicity can be defined based on the number of uses of axioms deemed to be deep or valuable, and prove a number of new results regarding the need to use at least three times some axioms in the proof of others. One such major example is Pappus implies Desargues, which is shown to require three uses of Pappus. A similar situation is encountered with Veblen's proof that the outer form of the Pasch axiom implies the inner form thereof. The outer form needs to be used at least three times in any such proof. We also mention the likely conflicting requirements of directness of a proof and the length of a proof. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

We present the largely twentieth century history of the discovery of the significance of Pappus and Desargues for the axiomatics of geometry. Their significance is followed in projective, affine, and orthogonality contexts. There is an extensive bibliography, that should allow the interested reader to take a comprehensive look at the research literature on these axioms.

We prove that Bachmann’s Lotschnittaxiom, which states that perpendiculars raised on the two legs of a right angle always meet, is equivalent, with respect to Hilbert’s plane absolute geometry, to the statement \(\mathbf{P}\): “For any point P, line l, with P not incident with l, and any line g, there exists a point G on g for which the distance to P exceeds the distance to l”.

Two results, proved synthetically in plane Euclidean geometry by Chisini in 1924—stating that: (i) if MAB is an isosceles triangle, with \({MA\equiv MB}\), inscribed in a circle \({\mathcal{C}}\), P1 and P2 are two points on \({{\mathcal C}}\) such that {B, Pi} separates {A, M} for \({i \in \{1,2}\}\), and {B, P2} separates {M, P1}, then \({AP_1+BP_1 < AP_2+BP_2}\), and (ii) of all triangles inscribed in a given circle the equilateral triangle has the greatest perimeter—are proved inside Hilbert’s absolute geometry.

The Erdős-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erdős-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erdős-Trost statement.

We present several formal theories for the arithmetic of the even and the odd, show that the irrationality of $\sqrt 2$ can be proved in one of them, that the proof must involve contradiction, and prove that the irrationality of $\sqrt {17}$ cannot be proved inside any formal theory of the even and the odd.

We prove that (i) a generalization of the Steiner-Lehmus theorem due to A. Henderson holds in Bachmann's standard ordered metric planes, (ii) that a variant of Steiner-Lehmus holds in all metric planes, and (iii) that the fact that a triangle with two congruent medians is isosceles holds in Hjelmslev planes without double incidences of characteristic $\neq 3$.

Two results are proved synthetically in Hilbert’s absolute geometry: (i) of all triangles inscribed in a circle, the equilateral one has the greatest area; (ii) of all triangles inscribed in a circle, the equilateral one has the greatest radius of the inscribed circle (which amounts, in the Euclidean case, to Euler’s inequality $$R\ge 2r$$ ).

By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.

Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples $(a,b,c)$ of positive integers for which $ab-c$ , $bc-a$ , and $ca-b$ are all powers of $2$ . We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.

In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in PA−+ Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in PA− together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.

We correct two typos and an error in The Arithmetic of the Even and the Odd, and provide an axiom system for a weak arithmetic in which the fact that a square root of an integer is either irrational or an integer can be proved.

In his 1854 Habilitationsvortrag Riemann presented a new concept of space endowed with a metric of great generality, which, through specification of the metric, gave rise to the spaces of constant curvature. In a different vein, yet with a similar aim, J. Hjelmslev, A. Schmidt, and F. Bachmann, introduced axiomatically a very general notion of plane geometry, which provides the foundation for the elementary versions of the geometries of spaces of constant curvature. We present a survey of these absolute geometric structures and their first-order axiomatizations, as well as of higher-dimensional variants thereof. In the 2-dimensional case, these structures were called metric planes by F. Bachmann, and they can be seen as the common substratum for the classical plane geometries: Euclidean, hyperbolic, and elliptic. They are endowed with a very general notion of orthogonality or reflection that can be specialized into that of the classical geometries by means of additional axioms. By looking at all the possible ways in which orthogonality can be introduced in terms of polarities, defined on (the intervals of a chain of subspaces of) projective spaces, one obtains a further generalization: the Cayley-Klein geometries. We present a survey of projective spaces endowed with an orthogonality and the associated Cayley-Klein geometries.

We ask if there is any right isosceles triangle in the hyperbolic plane, constructible with ruler and compass, for which the ratio of the hypotenuse to the side is rational. Although the question remains open, we determine that there are no such triangles with ratio $$1+\frac{m}{n}$$ , with $$n\le 12$$ .

The Pasch axiom P is shown to be equivalent, given the linear order axioms, to the conjunction of Pasch’s Theorem PT with the Weak Crossbar Theorem WCBT. Replacing P with PT and WCBT one gets a simplest axiom system for ordered planes. A likely missing link between PT and the outer form of the Pasch axiom OP, as well as an axiom system for the fragment of the geometry of ordered planes that can be expressed by at most 5 variables, when written in prenex form, are also determined. It is an open problem whether P can be proved inside the 5-variable fragment of plane ordered geometry.

The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its outer form with the statement that K 5 (or K 3,3) is not planar.

Axiom A16 from Pambuccian (Can. Math. Bull. 53, 534–541, 2010) is shown to be superfluous as it depends on axioms A1–A15. This provides a surprisingly simple axiom system in which the acute triangulation with seven triangles can be proved for any triangle, consisting only of A1–A15 in Pambuccian (Can. Math. Bull. 53, 534–541, 2010).

The theorem comparing the angles of two geodesic triangles with the same side lengths lying on surfaces with different curvatures, commonly attributed for the two-dimensional case to A.D. Alexandrov (1948) and for the n-dimensional Riemannian case to V.A. Toponogov (1959) , goes back, for the two-dimensional case, to Paolo Pizzetti (1907b) . Besides suggesting a correction of the historical narrative regarding the development of comparison geometry, the present note also mentions possible reasons for the oversight.

The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its inner form with a statement involving five individual variables.

We present a survey of the rich theory of betweenness and separation, from its beginning with Pasch’s 1882 Vorlesungen uber neuere Geometrie to the present.

We prove that, in ordered plane geometries endowed with a very weak notion of orthogonality, one can always triangulate any triangle into seven acute triangles, and, in case the given triangle is not acute, into no fewer than seven.

It is shown that a mapping \({\varphi: \mathfrak{A}\rightarrow \mathfrak{B}}\) between models \({\mathfrak{A}}\) and \({\mathfrak{B}}\) of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area and sharing a side into triangles that have the same property, must be a hyperbolic motion onto \({\varphi(\mathfrak{A})}\). The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for “the triangles abc and abd have the same area.”

A plane geometry is introduced, which although ordered such that both every line is densely linearly ordered without first or last element and such that every line partitions the points of the plane not incident with it into two convex half-planes, does not need to satisfy the Pasch axiom. An alternate axiomatization of ordered planes is obtained by adding several axioms, including the inner form of the Pasch axiom. The missing link between the inner form of Pasch’s axiom and the full Pasch axiom is herewith determined.

We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom (IP) does not imply its outer form (OP). We also show that OP can be properly split into IP and the weak Pasch axiom (WP) (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We prove that, in Hilbert’s plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that “If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well”, is equivalent to F. Bachmann’s Lotschnittaxiom, which states that “The perpendiculars on the sides a right angle always intersect.”

We show that the equivalence of the rectangle existence axiom to an axiom used by Maria Teresa Calapso to characterize the Euclidean metric can be proved inside rather weak, purely metric axiom systems for absolute planes.

We prove that, in plane absolute geometry, the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.

In this survey paper, we present several results linking quantifier-free axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions.

We formulate two remarkably elementary variants of theorems of the Alexandrov–Zeeman type, stated as definability statements, in minimalist axiomatic settings. The first is obtained by providing positive definitions for the notions of collinearity and segment congruence in terms of the notion of connectedness by a light ray, and is the logical counterpart of results stated in terms of mappings in [24] and [9]. The second is obtained by noticing that, in essence, results stating that mappings which preserve time-like lines preserve all lines, are the model-theoretic counterparts of definability results regarding the partial affine spaces introduced in [18].

It is shown that segment congruence can be defined by means of a positive existential definition in terms of segment inequality in several geometries over Archimedean ordered fields, and that it can be defined by means of a positive sentence in n-dimensional Euclidean geometry over arbitrary ordered fields.

We point out that the results in [12] are the model-theoretic counterpart of results established syntactically in [3] and [10], and that Martin’s theorem for the Euclidean 3- or higher-dimensional case, established in [5], does not depend on the Beckman-Quarles theorem, and can be rephrased as a result about axiomatizability and definability.

We show that the first-order theory of a large class of plane geometries and the first-order theory of their groups of motions, understood both as groups with a unary predicate singling out line-reflections, and as groups acting on sets, are mutually inter-pretable.

We present a definition for the co-punctuality of three lines in the hyperbolic plane in terms of line-orthogonality, correcting an omission in [2] and [1].

When we conjecture a theorem in geometry, we first try to prove it using whatever means we find at our disposal. After establishing its truth, we may follow it up by a reverse analysis, first described by Pappus of Alexandria as analysis, asking not whether the theorem in question holds, but what does one need to assume for it to hold (see [10] for the history of the regressive undertaking). Hilbert [4] expressed this enterprise with his characteristic eloquence