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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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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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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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 ...
44 views

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 ...
76 views

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. $A3$ ...
278 views

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 ...
185 views

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 ...
297 views

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 ...
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 ...