Questions tagged [axiomatic-geometry]

Questions about axiomatic systems for geometry. Use this tag if you're looking for a proof starting directly from some set of axioms (e.g., Hilbert's axioms for Euclidean geometry), or if you have a question about the axioms themselves.

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56 views

Is there an axiomatic foundation for n-dimensional Euclidean space?

In The Elements, Euclid establishes a set of axioms that dictate the behaviour of the plane. Hilbert refined these axioms to actually prove every proposition of The Elements formally. In Hartshorne'...
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Fundamental questions on "The shortest distance between two points is a straight line."

In retrospective of history of mathematics, I am trying to reconstruct the answers to following fundamental questions: Who proved first in a certain geometry that "The shortest distance between ...
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Tarski's elementary Euclidean geometry

It is a well-known theorem of Tarski that (what is now called) Tarski's elementary Euclidean geometry is a decidable theory. This is a first-order theory, as opposed to Hilbert's second-order ...
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Constructing a line through $P$ that meets $\overline{AB}$ and $\overline{AC}$ in points $D$ and $E$ such that $\overline{DB}\cong\overline{EC}$ [duplicate]

Geometry problem in Hironaka Heisuke's book. (I don't think this book is translated into English.) I saw this beautiful problem which the author said he solved it in high school: $P$ is a point ...
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Calculating angles of diagonal lines in quadrangle, using axiomatic geometry

In my previous question: Common rules in calculating angles of diagonal lines in quadrangle, using synthetic(axiomatic) geometry. I wanted to know the common rule for calculating angles of diagonal ...
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128 views

Common rules in calculating angles of diagonal lines in quadrangle, using synthetic(axiomatic) geometry.

In this quadrangle, if $x, y, a, b$ is determined, $c$ has to be determined too. Can $c$ be expressed using $x, y, a, b$? Searching similar questions in axiomatic geometry is hard. If similar question ...
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Constructing an equilateral triangle using Tarski's axioms for geometry

In Euclid's first geometry proposition, he constructs an equilateral triangle given an arbitrary line segment. I was wondering if it was possible to prove this straight from Tarski's axioms for ...
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Prove that a disk is convex in Tarski's geometry

Is it possible to prove that a disk is convex using only Tarski's axioms of geometry? Specifically, I'm looking for a proof of: $\forall a\,\forall b\,\forall c\,\forall d\,\forall e\,[ ab \equiv ac \...
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57 views

Can the study of conics/quadrics be done axiomatically?

Using Tarski's axioms or Hilbert's axioms, Euclidean geometry can be described synthetically (in a way that can even be formalized in Coq), i.e. within a theory of first-order logic, or second-order ...
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Collinearity Criterion in euclidean geometry

Problem setting Let $(M,d)$ be a metric space. We will say that $l\subset M$ is a line if there is a surjective function $f:\mathbb{R}\to l$ such that $d(f(x),f(y))=|x-y|$. If $l$ is a line, being $f$ ...
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Removing the protractor and SAS postulates

I am reading the book "Foundations of Geometry" by Gerard A. Venema (2nd ed). A draft of the axiomatic setting of the book is the following (I will expand if required): Incidence postulate: ...
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What does it mean for one geometrical axiom to be considered _equivalent_ to another geometrical axiom?

What does it mean for one geometrical axiom to be considered equivalent to another geometrical axiom? For example consider Playfair`s axiom: In a plane, given a line and a point not on it, at most one ...
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For Desarguian projective planes, coordinatization is inverse to the $K \mapsto K\mathbf{P}^2$ construction

In J. C. Baez's February 9, 2000 entry on his website it is claimed that the process of coordinatizing a Desarguian projective plane (i.e. obtaining a skew field/division ring in the style of the ...
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Is "There exists a circle passing through any three noncollinear points" really equivalent to Euclid's parallel postulate?

This site https://www.ics.uci.edu/~eppstein/junkyard/parallel-postulate.html lists from the book "The Foundations of Geometry and the Non-Euclidean Plane" by George E. Martin 26 propositions ...
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Synthetic geometry theorems that relate lengths to areas

Does anyone know any synthetic geometry theorems (so, no algebra at all) or sources with synthetic geometry theorems, that relate lengths to areas? My only reference currently is Euclid's Elements, ...
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Existence of skew lines in 3D

I need to prove existence of skew lines in three dimensional space? I'm looking for starting point. Is it possible to prove only using axioms of geometry?
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Are the theories of betweenness the same in $\mathbb{R}^n$ for all $n\geq 2$?

Consider $\mathbb{R}^n$, for some $n$ greater than or equal to $2$. We can form a structure by adjoining to it the ternary betweenness relation $B(x,y,z)$. Are all those structures elementarily ...
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Prove a certain quadrilateral is actually a Saccheri Quadrilateral

Question: "Show that a quadrilateral ABCD, which has angle C = angle D = right angle; and angle A is congruence to angle B, is a Saccheri Quadrilateral." My attempt: As by definition, a ...
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How first-order logic based ZFC can model mathematical theories formulated with higher order logic? [duplicate]

Zermelo–Fraenkel set theory with axiom of choice (ZFC) can be used as a foundation of mathematics. It means, that any theory can be expressed in terms of ZFC. In particular, it should be possible to ...
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86 views

What logic can be used express Hilbert's axiomatization of Euclidian geometry?

Can the first order logic be used to express Hilbert axioms for Euclidean geometry as well as all theorems that follow from them? If it is not the case, what logic can be used? Does it have a name? In ...
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Proving (via Hilbert's axioms) that, when two parallel lines are cut by a third line, they make congruent angles.

Prove that, when two parallel lines are cut by a third line, they make congruent angles. I'm not using Euclid's axioms, but instead I'm using Hilbert's. This is Theorem 19 of Hilbert's "The ...
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Can one prove non-trivial congruences of triangles without SAS or other congruence axioms?

In math class, I was told we need to take SAS as an axiom, otherwise we could not prove any congruences besides a triangle and itself. Is that really true? Is there a model of Hilbert's Euclidean ...
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Prove: "if three points are on a straight line, at least one point is between the other two."

In Wikipedia, the third order axiom of Hilbert's axioms states that "Of any three points situated on a line, there is no more than one which lies between the other two. Note: The existence part ...
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Catergorical Axiomatic System

There are two axioms: There are two different points $A, B$, and two different lines $\ell_1, \ell_2$ in such way that A, B are located on both of this lines $(A,B \in \ell_1$ and $A,B \in \ell_2)$. ...
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Defining real numbers through geometry

Following this paper, we can define a straight line as a triplet $(L,\mathcal{B},\cong)$ were $L$ is a set (the set of points in the line), $\mathcal{B}$ is a ternary relation over $L$ (called the ...
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Do Tarski's (geometry) axioms imply that all zero segments are congruent?

Tarski's axioms are an alternate formalization of geometry (similar to axiom sets of Euclid and later Hilbert). Do these axioms imply: $$\forall\; x,y\in \text{points},\; x x\equiv y y?$$ If yes, ...
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Perpendicular lines with synthetic geometry and minimal assumptions (axioms)

I'm trying to understand Euclidean geometry the hard way. I don't want to start with analytical geometry, building on coordinates and vector spaces, nor on the Euclidean approach of synthetic geometry ...
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Definition of the sinusoid in terms of differential equation

Let us define $\sin(x)$ as the solution to $$y'' = -y$$ with initial conditions $y(0) = 0$ and $y'(0) = 1$. From this definition, we should be able to deduce from first principles that $\sin(x)$ is ...
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Proving Parallel Postulate by showing that it is undecidable

In the Numberphile video "Gödel's Incompleteness Theorem" (via YouTube), Professor Marcus Du Sautoy mentions that the Riemann hypothesis could be proved true by proving that it was undecidable, ...
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What motivated people like Euclid to create such a general axiomatic systems like the one in the book Elements so early in history?

I think I understand why people wanted (and still wants of course) to prove some mathematical statements. Example of that would be proof of Pythagorean theorem. People noticed earlier than Pythagoras ...
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Parallellogram in Hyperbolic Geometry is not composed of two congruent triangles.

I am trying to show that in hyperbolic geometry, any diagonal of a parallellogram $\square ABCD$ divides the parallellogram into two non-congruent triangles. I have tried assuming the contrary, and ...
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Boundary lines in Euclid geometry.

I have noted that by many plane geometric proofs or constructions the following propositon is implicitly used: Proposition: Straight line and circle are boundary lines. where the boundary line is ...
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Line intersecting three parallel lines, in neutral geometry

The following theorem is definitely true in Euclidean geometry: Theorem. Given any three parallel (that is, pairwise non-intersecting) lines $\ell_1, \ell_2, \ell_3$, there is a fourth line which ...
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221 views

Converse of the Crossbar Thereom

I need help proving the converse of the crossbar theorem. The crossbar theorem states that Let $\angle{ABC}$ be an angle and $D \in Int(\angle{ABC})$. Then $\overrightarrow{BD} \cap Int(\overline{AC})...
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Book recommendation for Axiomatic three-dimensional geometry and analytic geometry

I'm reading John Lee's book Axiomatic Geometry and I enjoy it a lot. It includes a detailed treatment of Euclidean plane geometry with rigorous proofs from axioms. I'm looking for books about ...
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Decomposition of a plane

Angular line is a union of two half-lines with the same origin in the plane. Open angle is the set of all points which are on the same side of the angular line (they are on the same side if there is a ...
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Orientation on a line and equivalence relation

In order to define orientation on a line, my textbook defines half-line as the set of points that are on the same side of a given point on the line. Then, they define the relation of "pointing in the ...
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How do I affirm that a triangle defines a plane?

I am trying to postulate the Euclidean plane surface. I have postulates of the straight line and of extending a straight line. I want to fill the gap between a triangle and a plane. How does a ...
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Using another congruence criterion as axiom in Hilbert's axiomatic geometry

I have a question about Hilbert's axiomatic system. In this system we have the SAS (Side-Angle-Side) congruence criterion as an axiom, and then we prove the SSS (Side-Side-Side) and ASA (Angle-Side-...
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Importance of Axiomatic Geometry

What is the importance of axiomatic geometry? I understand that analytic geometry is really helpful for some kinds of geometries. And in most instances it's all we need. Is the axiomatic approach ...
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Existence of Right Angle in Hilbert Axioms

Hilbert, in Foundations of Geometry briefly mentions that the existence of right angles is a corollary to the supplementary angle theorem. (i.e. If two angles are congruent, then their supplementary ...
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Conditions for a dual geometry to be a model of incidence geometry

I study Euclidean geometry I know, models of incidence geometry must satisfy 3 axioms: $A1$ Two distinct points belong to exactly one line. $A2$ There are at least 2 points on every line. $A3$ ...
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Ordering angles

I have recently been introduced to Hilbert's axioms of geometry. Right now I am studying angles. A text I have been using as a guide defines an angle and its opening in the following way: An angle is ...
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Axioms of order in geometry and ordered fields

I am considering axioms of incidence and axioms of order for plane geometry by Hilbert: I1: Two points determine the unique line. I2: Each line contains (at least) two points. I3: There are three ...
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How much of Euclid's number theory can be reproduced in Hilbert's axiomatization of geometry?

In book VII of the Elements, Euclid develops some fundamental results from number theory. How much of this, if any, can be reproduced in Hilbert's axiomatization? It seems to me that the answer is ...
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Model of ordered plane with continuity and at least one line not isomorphic to $\mathbb{R}$.

Background information Among primitive notions in mamy axiomatic systems for geometry we have ternary betweenness relation. Let's look at the following axioms of strict one-dimensional betweenness: ...
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Continuity axioms for the real projective line in terms of a separation relation

I'm interested in (categorical) axiomatization of the real projective line in terms of a quaternary separation relation. First I'll give some axioms which describe the separation relation i.e. the ...
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If two medians are congruent... is the triangle isosceles in a Hilbert plane?

If $ABC$ is a triangle for which two medians are congruent... is it true that the triangle $ABC$ is isosceles in a general Hilbert plane? I am having a little bit of trouble trying to prove this, if ...
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398 views

Are the following statements equivalent to the parallel postulate?

In one of my Elementary Geometry previous exams, one of the questions was the following: Study if the following statements are equivalent to the paralellism axiom: $(i)$ Any three straight lines have ...
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Reference for the notion of "dimension" in incidence geometry?

Consider an incidence geometry as in an elementary course on geometry. The german wikipedia https://de.wikipedia.org/wiki/Inzidenzgeometrie mentions a well defined notion of "dimension". Does ...