Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Euclidean Geometry

Field: Geometry

The study of plane and solid figures based on Euclid's axioms.

Sequence of Expressions

Axiom

Axioms

Let M=(S,Inc,Bet,Dist)\mathcal{M} = (S, \text{Inc}, \text{Bet}, \text{Dist}) be a mathematical structure, where SS is the set of points, Inc⊆S×S\text{Inc} \subseteq S \times S is the incidence relation, Bet\text{Bet} is the betweenness relation, and Dist:S×S→R≥0\text{Dist}: S \times S \to \mathbb{R}_{\ge 0} is a metric function. The system of Euclidean Axioms AE\mathcal{A}_{E} is the set of first-order logical statements that define the properties of M\mathcal{M}. These axioms include:\\ \text{Ax}_1(Incidence): (Incidence): \forall A, B \in S, (A \text{ Inc } B) \iff (A = B)or or (A \text{ and } B \text{ are distinct points}).Ax2.\\\text{Ax}_2 (Betweenness): ∀A,B,C∈S\forall A, B, C \in S, if BB lies between AA and CC, then Dist(A,C)=Dist(A,B)+Dist(B,C)\text{Dist}(A, C) = \text{Dist}(A, B) + \text{Dist}(B, C).\\\text{Ax}_3(Congruence):Theexistenceofanisometrygroup (Congruence): The existence of an isometry group \text{Isom}(S)suchthatforanytwosegments such that for any two segments \overline{AB}and and \overline{CD},, \text{Dist}(A, B) = \text{Dist}(C, D)impliesarigidtransformationmappingonetotheother.Ax4 implies a rigid transformation mapping one to the other.\\\text{Ax}_4 (Euclidean Parallel Postulate): For any line LL and any point P∉LP \notin L, there exists a unique line L′⊂SL' \subset S such that L′L' passes through PP and L′L' is parallel to LL (i.e., L′∩L=∅L' \cap L = \emptyset and L′L' maintains a constant distance from LL).\\\text{Ax}_5(Completeness/Dimension):Thespace (Completeness/Dimension): The space Sisacomplete,connected,andsimplyconnectedmanifoldofdimension2,satisfyingthemetricpropertiesof is a complete, connected, and simply connected manifold of dimension 2, satisfying the metric properties of \mathbb{R}^2$.
For a right-angled triangle with legs a,ba, b and hypotenuse cc, a2+b2=c2a^2 + b^2 = c^2.
If A,B,CA, B, C are points on a circle where ACAC is a diameter, then the angle ∠ABC\angle ABC is a right angle.
Geometry is the science of correct reasoning on incorrect figures. — George Pólya, How to Solve It, p. 208 - Euclid's axioms: In his dissertation to Trinity College, Cambridge, Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time. It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the parallel postulate was not necessarily valid and its applicability was an empirical matter, deciding whether the applicable geometry was Euclidean or non-Euclidean. - Hilbert's axioms: Hilbert's axioms had the goal of identifying a simple and complete set of independent axioms from which the most important geometric theorems could be deduced. The outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel postulate. - Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. This system relies heavily on the properties of the real numbers. The notions of angle and distance become primitive concepts. - Tarski's axioms: Alfred Tarski (1902–1983) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets. Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain sense: there is an algorithm that, for every proposition, can be shown either true or false. (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.) This is equivalent to the decidability of real closed fields, of which elementary Euclidean geometry is a model. - ^Bertrand Russell (1897). "Introduction". An essay on the foundations of geometry. Cambridge University Press. - ^George David Birkhoff; Ralph Beatley (1999). "Chapter 2: The five fundamental principles". Basic Geometry (3rd ed.). AMS Bookstore. pp. 38 ff. ISBN0-8218-2101-6. - ^James T. Smith (10 January 2000). "Chapter 3: Elementary Euclidean Geometry". Cited work. John Wiley & Sons. pp. 84 ff. ISBN9780471251835. - ^Edwin E. Moise (1990). Elementary geometry from an advanced standpoint (3rd ed.). Addison–Wesley. ISBN0-201-50867-2. - ^John R. Silvester (2001). "§1.4 Hilbert and Birkhoff". Geometry: ancient and modern. Oxford University Press. ISBN0-19-850825-5. - ^Alfred Tarski (2007). "What is elementary geometry". In Leon Henkin; Patrick Suppes; Alfred Tarski (eds.). Studies in Logic and the Foundations of Mathematics – The Axiomatic Method with Special Reference to Geometry and Physics (Proceedings of International Symposium at Berkeley 1957–8; Reprint ed.). Brouwer Press. p. 16. ISBN978-1-4067-5355-4. We regard as elementary that part of Euclidean geometry which can be formulated and established without the help of any set-theoretical devices - ^Keith Simmons (2009). "Tarski's logic". In Dov M. Gabbay; John Woods (eds.). Logic from Russell to Church. Elsevier. p. 574. ISBN978-0-444-51620-6. - ^Cite error: The named reference Tarski 1951 was invoked but never defined (see the help page). - ^Franzén, Torkel (2005). Gödel's Theorem: An Incomplete Guide to its Use and Abuse. AK Peters. ISBN1-56881-238-8. Pp. 25–26.
- Angle bisector theorem - Butterfly theorem - Ceva's theorem - Heron's formula - Menelaus' theorem - Nine-point circle - Pythagorean theorem
Let (M,g)(M, g) be a smooth, connected, nn-dimensional Riemannian manifold. The geometry defined by (M,g)(M, g) is Euclidean if and only if the sectional curvature KK of the manifold is identically zero everywhere. Formally, this means that for any point p∈Mp \in M and any two-dimensional subspace Π⊂TpM\Pi \subset T_p M (the tangent space at pp), the sectional curvature K(Π)K(\Pi) must satisfy:\n\nK(Π)=R(v,w,v,w)g(v,v)g(w,w)−g(v,w)2=0K(\Pi) = \frac{R(v, w, v, w)}{g(v, v)g(w, w) - g(v, w)^2} = 0\n\nwhere RR is the Riemann curvature tensor, and v,wv, w are tangent vectors spanning Π\Pi. Furthermore, the manifold (M,g)(M, g) must be locally isometric to the standard Euclidean space (Rn,gEuc)(\mathbb{R}^n, g_{Euc}), where gEucg_{Euc} is the standard Euclidean metric tensor δij\delta_{ij}. This condition ensures that the parallel postulate holds in the strongest sense.
Beginner
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.\na2+b2=c2a^2 + b^2 = c^2