What is Euclidean geometry: Definition and 48 Discussions

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas.

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  1. Mark Harder

    I Chiral symmetries in E[SUP]^n[/SUP]

    As a biochemist, I deal with chirality of molecules all the time. If you have a tetrahedral molecule, for example a carbon atom, and all 4 vertices are labeled differently, as in different atoms on each one, then that molecule has a mirror-symmetric one that cannot be superimposed on the...
  2. VladStoyanoff

    I Given a triangle ABC, on which side of the triangle does the point D lie?

    So I've got the following problem: I have points A, B, and C which form a triangle in a 3D space (each point of the triangle has x,y, and z coordinates). I need to find out on which side of the triangle point D lies. I do not have access to the normal of the triangle. How am I supposed to...
  3. S

    MHB Euclidean Geometry - Demonstration Exercise

    (a) Let be m a line and the only two semiplans determined by m. (i) Show that: If are points that do not belong to such , so and are in opposite sides of m. (ii) In the same conditions of the last item, show: and . (iii) Determine the union result , carefully justifying your answer...
  4. K

    I Calculations to prove the non-Euclidean nature of 3-D space

    In Anthony French's book, Newtonian Mechanics, while explaining the non-Euclidean nature of the 3-d space, he poses a problem (I have rephrased it slightly): Suppose you are on Earth's equator (r = 6,400 km) at the prime meridian (point I). You first walk along the equator 1000 miles east and...
  5. greg_rack

    Euclidean geometry: main theorems, formulas and concepts

    Hi guys, Hopefully, no geometry-enthusiasts are going to read these next few lines, but if that's the case, be lenient :) I have always hated high-school geometry, those basic boring theorems about triangles, polygons, circles, and so on, and I have always "skipped" such classes, studying...
  6. T

    I Does Euclidean geometry require initial fine-tuning?

    https://en.wikipedia.org/wiki/Flatness_problem The flatness problem (also known as the oldness problem) is a cosmological fine-tuning problem within the Big Bang model of the universe. The fine-tuning problem of the last century was solved by introducing the theory of inflation which flattens...
  7. LittleRookie

    Two geometry problems -- lines and straightedges

    Hello all, I need some help on two exercises from Kiselev's geometry, about straight lines. Ex 7: Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight. Hint: Flip the straightedge upside down. I would...
  8. S

    A Is there any theory that can be modeled in any type of space?

    Is there any theory in physics that can be modeled in any type of space (Hilbert space, Euclidean, Non-Euclidean...etc)? And if yes, could that theory also contain/be compatible with all types of (physical) symmetries?
  9. cianfa72

    I Difference (if any) between E^1 x E^3 and E^4

    Hi, reading this link galilean spacetime a doubt raised to me: which is the difference (if any) between ##\mathbf E^1\times\mathbf E^3## and ##\mathbf E^4## ? I believe each space there (spaces involved in the cartesian product too) has to be regarded as equipped with standard Euclidean structure
  10. E

    A Vec norm in polar coordinates differs from norm in Cartesian coordinates

    I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
  11. shintashi

    I Equidistance of Points in a Sphere?

    I've been trying to wrap my head around equidistant points, like platonic solid vertices inside a sphere where the points touch the sphere surface. This led me to the strange and unusual world of mathematical degeneracy, henagons, dihedrons, and so on, along with the lingering question of...
  12. Vicol

    A Let's remove one axiom from Euclidean geometry

    I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?
  13. parshyaa

    I Question about Euclid's parallel postulate (5th postulate).

    Why can't we prove euclids fifth postulate What's wrong in this proof: why can't we prove that there is only one line which passes through a single point which is parallel to a line. If we can prove that two lines are parallel by proving that the alternate angles of a transverse passing...
  14. N

    Euclidean and non Euclidean geometries problems

    So I was reading this book, "Euclidean and non Euclidean geometries" by Greenberg I solved the first problems of the first chapter, and I would like to verify my solutions 1. Homework Statement Homework Equations [/B] Um, none that I can think of? The Attempt at a Solution (1) Correct...
  15. Ameer Bux

    Can someone help me with these 2 Euclidean geometry questions?

    Homework Statement write the proof Homework Equations none The Attempt at a Solution I've tried 5 times, got nowhere
  16. micromass

    Schools In High School and Want to Do Advanced Mathematics? - Comments

    micromass submitted a new PF Insights post In High School and Want to Do Advanced Mathematics? https://www.physicsforums.com/insights/wp-content/uploads/2016/03/high school-math.png [Broken] Continue reading the Original PF Insights Post.
  17. Jianphys17

    Prerequisites for non Euclidean geometry

    Hi, i would be very interested to start learning hyperbolic geometry, what would be the necessary prerequisites to begin it's study? :smile:
  18. P

    Two dimensional Lorentzian vs Euclidean geometry

    Reading a somewhat long and argumentative thread here inspired the following unrelated question in my mind: Where does a 2 dimensional flat Lorentzian geometry depart from Euclidean geometry as axiomatized by Euclid? I.e. Euclid's axioms (in modern language) can be taken to be: We can...
  19. Odious Suspect

    Geometry Seeking concise review of Elementary Euclidean Geometry

    I'm seeing a presentation of Euclidean geometry that isn't hand-holdy. I've looked at some textbooks used in high schools these days, and it's hard to find the axioms and theorems in the midst of all the condescension. I just want something that states the definitions, axioms and basic...
  20. Mastermind01

    Need Help With Plane Geometry? Learn More Here!

    Hello, I am totally bad at geometry , by geometry I mean plane euclidean geometry with similarities and circles. I sometimes feel totally lost with problems. For example: The parallel sides of trapezoid ABCD are 3 cm and 9 cm(AB and DC).The non parallel sides are 4 cm and 6 cm(AD and BC).A...
  21. modularmonads

    [Euclidean Geometry] Kiselev's Plainimetry Question 242

    Homework Statement Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2)...
  22. astrololo

    (euclidean geometry) Euclid's elements proposition 13 book

    "Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd. Therefore a circle does not touch a circle...
  23. Ahmed Abdullah

    Smallest infinity for Euclidean geometry to work

    If we choose rational numbers to represent points on a line then there will be gaps on the line and consequently the plane will be full of holes. Then we cannot say that two non-parallel line must intersect on a point (because they may meet at the gaps). So obviously we need point arranged more...
  24. N

    How do axioms for Euclidean geometry exclude non-trivial topology?

    Think for example of the torus as a square with the proper edges identified. Viewed like this (i.e. using the flat metric), it clearly has zero curvature everywhere. More specifically, it seems Euclid's axioms are satisfied. But however we have non-trivial topology. So what's up? Or is...
  25. V

    Euclidean geometry: aadvanced rider

    Homework Statement LPN is a tangent to circle ADP. Circle BCP touches the larger circle internally at P. Chord AD cuts the smaller circle at B and C and BP and CP are joined Homework Equations The Attempt at a Solution ∠P4+5 = ∠B1 (tan chord theorem) ∠P1+2 = C1 (tan chord theorem)
  26. M

    All Triangles are Isosceles Proof (Euclidean Geometry)

    Prove that in any triangle, if the angle bisectors of two angles are congruent, then the triangle is isosceles Before I give my proof, here is a lemma to it: If a pair of vertical angles both have angle bisectors, then all resulting angles are congruent. Given: Vertical Angles ∠2 and ∠4, and...
  27. T

    When will euclidean geometry become ?

    Hello everybody, Based of some information that I recently learnt(which I don't know if they are right or wrong), I start asking myself this question about the euclidean geometry. Ok, this geometry is basically founded on straight lines, and what I have learned is there is no such a thing as a...
  28. T

    Is it necessary to study Euclidean Geometry before Differential Geom.?

    Hi, I'm a Physics undergraduate, and this semester I have the option to choose between Geometry (Axiomatic Euclidean Geometry) and other disciplines. In the next year I want to be ready to study Differential Geometry, but I don't know if I need to study Euclidean Geometry first. The teacher of...
  29. N

    Projectile motion equations using Euclidean Geometry only

    Here is a spreadsheet I made to solve projectile motion problems. It uses only euclidean geometry to solve for the answers, but it gives the answers as slopes, not angles. All variables entered also must be in slopes and not angles. I added a section that does the conversion between the two...
  30. G

    Euclidean geometry and complex plane

    Can someone please describe to me how Euclidean Geometry is connected to the complex plane? Angles preservations, distance, Mobius Transformations, isometries, anything would be nice. Also, how can hyperbolic geometry be described with complex numbers?
  31. S

    Inequality with Circle and Triangle in Euclidean Geometry

    Homework Statement Please see below... Homework Equations Please see below... The Attempt at a Solution Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution... Thank you...
  32. K

    Euclidean geometry doesn't exist?

    As a newbie, I apologize if this topic has been discussed before. It seems to me that one result of quantum physics is that Euclidean geometry is artificial and cannot be represented in real space. For example, there can be no such thing as a straight line in granular quantum space. And...
  33. A

    Maths Project, Euclidean Geometry

    1. Maths project to investigate compass and straightedge constructions 2. Most of the project is fine, but i need to find out the mimimum number of constructions to bisect an angle, a line segment, etc. 3. I can prove that you can bisect an angle, and it requires 4 steps to do it...
  34. A

    How is the 3-d space an approximation of Euclidean Geometry?

    I would like to know the basic experimental observations or the logic which prove that the 3-d space which we inhabit is a close approximation of Euclidean Geometry. is it because parallel lines don't appear to converge or diverge? But how is this established, as we can't draw perfect straight...
  35. M

    Euclidean geometry proof concerning circles

    i really need help with this proof. suppose two circles intersect at points P and Q. Prove that the line containing the centers of the circles is perpendicular to line segment PQ
  36. X

    Why Euclidean Geometry Fails in Real-World Space

    Considering that space is not curved or warped (as some pop books will falsely lead you to believe) why is that Euclidean Geometry is not true in the real world? I mean light bends in space because it falls in a gravitational field like everything else (because it has energy which is...
  37. D

    Euclidean Geometry: 8.2.1 & 8.2.2 Solutions

    [b]1. Homework Statement http://img195.imageshack.us/img195/5122/200282.gif [Broken] Homework Equations The Attempt at a Solution 8.2.1) Let D1 = x D4=D1 =x D4=L1 (tan chord theorem) L1=x D1=L2 L2=x angle KLM=2x KNM=2x(opp angles in //gram)...
  38. J

    Incompleteness of Euclidean Geometry: Proving the Parallel Postulate

    "For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms." http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems The parallel postulate says that, if a line segment...
  39. C

    College Euclidean Geometry

    I have this question: Inside a square ABDE, take a point C so that CDE is an isosceles triangle with angles 15 degrees at D and E. What kind of triangle is ABC? I put C close to the bottom to get my isosceles triangle. According to the answer in back, the triangle ABC is equilateral...
  40. H

    Elementary Algebra & Euclidean Geometry

    I would say by now, I'm an expert in manipulating equations and playing with algebra. However, I've also realized I have no idea why some of the operations I do are valid. For example... why is (x+2)(x-2) = x^2 - 4? Why does this expansion work? I'm guessing it preserves some kind of field...
  41. M

    Does this forum include euclidean geometry?

    I am teaching euclidean geometry this fall and realized i don't know it that well. there are some famous modern versions of the axioms which do not completely satisfy me, such as hilberts, gasp. i said it. i especially like the new book by hartshorne, geometry euclid and beyond, because he...
  42. R

    Euclidean geometry

    I wanted to know if euclidean geometry is to do with the real world ? generalisations of vector space to anything that satisfies the axioms for a vector space can be made, but how can geometry be studied without reference to the real world ? roger
  43. E

    Surface Volume in 4-d graph: Euclidean Geometry Question

    "Surface Volume" in 4-d graph: Euclidean Geometry Question Suppose you have a smooth parametrically defined volume V givin by the following equation. f(x,y,z,w)= r(u,s,v) = x(u,v,s)i + y(u,v,s)j +z(u,v,s)k + w(u,v,s)l Consider the vectors ru=dr/du, where dr/du is the partial...
  44. E

    Euclidean Geometry Question

    Suppose you have a smooth parametrically defined volume V givin by the following equation. f(x,y,z,w)= r(u,s,v) = x(u,v,s)i + y(u,v,s)j +z(u,v,s)k + w(u,v,s)l Consider the vectors ru=dr/du, where dr/du is the partial derivitive of r with respect to the parameter u. Similarly, rv =...
  45. I

    Proving AB is Not Equal to CD in Euclidean Geometry

    If you had line AB is parallel to BC and BC is parallel to CD, is AB parallel to CD? ----> Not if AB=CD since a line (at least in Euclidean Geometry) cannot be parallel to itself. How would you prove that AB is not line CD? PLEASE NOTE: Base all your input in the realm of Euclidean...
  46. T

    Is Non-Euclidean Geometry the Key to Unlocking New Mathematical Discoveries?

    Hi everyone, i have to do a general math project for my math course. It is nothng special, just a proof of a theorem on my choice and a bit of history and interesting facts. I kind of decided to do it on non-euclidean geometry, because it is fun. Now my question: does anybody know a good...
  47. S

    Proving Bisection of Angles with Euclidean Geometry

    I was at brunch this morning and I met a young man of about 35 years who is a musician but studied mathematics in college. He mentioned to me that one of the great problems of mathematics is the problem of trisecting the angle. He taught me how to bisect an angle, and kept emphasizing that...