Grundlagen der Geometrie

en · unofficial · sha256:72129956c128c360ec86178c3978e4eb7ef3344162da2cddf1510ca717fb4dc4

III. In a plane α there can be drawn through any point A, lying outside of a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A. This statement of the axiom of parallels contains two assertions. The first of these is that, in the plane α, there is always a straight line passing through A which does not intersect the given line a. The second states that only one such line is possible. The latter of these statements is the essential one, and it may also be expressed as follows:

Referencing norms: AxiomIIIPlayfair, Parallele, ParalleleDurch, PlayfairParallele, Schneidet, parallele, parallele_durch, playfair_parallele, schneidet, HilbertReading, HilbertReading/HilbertBaseChosen

en · unofficial · sha256:582e26c4531c73d0d526f4f7196035a78abfd23d9a800e9c0dd52db5fce41249

II, 1. If A, B, C are points of a straight line and B lies between A and C, then B lies also between C and A.

Referencing norms: AxiomII1, AxiomII1Symmetrie, umgekehrte_lage, zwischen

en · unofficial · sha256:2cde85f0b8af176693672d9cb0c7b8cdc4d199caab41379981f27715afdda675

II, 2. If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.

Referencing norms: AxiomII2Fortsetzung, AxiomII2Innen, Fortsetzungspunkt, InnererPunkt, fortsetzungspunkt, innerer_punkt

en · unofficial · sha256:5c0bc3a89105ff53813ce08b21e292775d4512fa793e3fdcef6d06f364860e48

II, 3. Of any three points situated on a straight line, there is always one and only one which lies between the other two.

Referencing norms: AxiomII3EindeutigZweiteDritte, AxiomII3EindeutigZweiteErste, AxiomII3Existenz, LageBestimmtDritte, LageBestimmtErste, LageBestimmtZweite, lage_bestimmt

en · unofficial · sha256:0641f729844fdd148c13bf9cb03110d3cb6faa8bc6f04495076c556dbf5a63e0

II, 4. Any four points A, B, C, D of a straight line can always be so arranged that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D. Definition. We will call the system of two points A and B, lying upon a straight line, a segment and denote it by AB or BA. The points lying between A and B are called the points of the segment AB or the points lying within the segment AB. All other points of the straight line are referred to as the points lying outside the segment AB. The points A and B are called the extremities of the segment AB.

Referencing norms: Anordenbar01, Anordenbar02, Anordenbar03, Anordenbar04, Anordenbar05, Anordenbar06, Anordenbar07, Anordenbar08, Anordenbar09, Anordenbar10, Anordenbar11, Anordenbar12, Anordenbar13, Anordenbar14, Anordenbar15, Anordenbar16, Anordenbar17, Anordenbar18, Anordenbar19, Anordenbar20, Anordenbar21, Anordenbar22, Anordenbar23, Anordenbar24, AxiomII4, GeordneteVierergruppe, anordenbar, geordnete_vierergruppe

en · unofficial · sha256:5c198918f121fa7dc8039e9e4b5102e770f7a7e528c3b1539687bf3f29ccf2ba

II, 5. Let A, B, C be three points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the straight line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC. Axioms II, 1–4 contain statements concerning the points of a straight line only, and, hence, we will call them the linear axioms of group II. Axiom II, 5 relates to the elements of plane geometry and, consequently, shall be called the plane axiom of group II.

Referencing norms: AustrittUeberAC, AustrittUeberBC, AxiomII5Pasch, GeradeInDreiecksebene, Meidet, StreckeGetroffen, austritt, dreieck, gerade_in_dreiecksebene, meidet, strecke_getroffen

en · unofficial · sha256:43ce7e45cc51047756e93b959fb81aa181e214787362da11917d348cd35e38df

IV, I. If A, B are two points on a straight line a, and if A' is a point upon the same or another straight line a', then, upon a given side of A' on the straight line a', we can always find one and only one point B' so that the segment AB (or BA) is congruent to the segment A'B'. We indicate this relation by writing AB≡ A'B'. Every segment is congruent to itself; that is, we always have AB≡ AB. We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in one and and only one way.

en · unofficial · sha256:d8e4b26f58cb3f6c296a87617efa728da934bc35bf8bdadcf51e7a02cd86c6c3

IV, 2. If a segment AB is congruent to the segment A'B' and also to the segment A”B”, then the segment A'B' is congruent to the segment A”B”; that is, if AB ≡ A'B and AB ≡ A”B”, then A'B' ≡ A”B”.

en · unofficial · sha256:c0bef09ea7d2787224efebac615a7671413df183608aeccca7c988d196d4e8f5

IV, 3. Let AB and BC be two segments of a straight line a which have no points in common aside from the point B, and, furthermore, let A'B' and B'C' be two segments of the same or of another straight line a' having, likewise, no point other than B' in common. Then, if AB ≡ A'B' and BC ≡ B'C', we have AC ≡ A'C'. Definitions. Let α be any arbitrary plane and h, k any two distinct half-rays lying in α and emanating from the point O so as to form a part of two different straight lines. We call the system formed by these two half-rays h, k an angle and represent it by the symbol ∠(h, k) or ∠(k, h). From axioms II, 1–5, it follows readily that the half-rays h and k, taken together with the point O, divide the remaining points of the plane a into two regions having the following property: If A is a point of one region and B a point of the other, then every broken line joining A and B either passes through O or has a point in common with one of the half-rays h, k. If, however, A, A' both lie within the same region, then it is always possible to join these two points by a broken line which neither passes through O nor has a point in common with either of the half-rays h, k. One of these two regions is distinguished from the other in that the segment joining any two points of this region lies entirely within the region. The region so characterised is called the interior of the angle (h,k). To distinguish the other region from this, we call it the exterior of the angle (h,k). The half rays h and k are called the sides of the angle, and the point O is called the vertex of the angle.

en · unofficial · sha256:0ee149aba3f9436590e4eb2457ec4c11817d32a46c181105894fba4627d5dffd

IV, 4. Let an angle (h,k) be given in the plane α and let a straight line a' be given in a plane α'. Suppose also that, in the plane α', a definite side of the straight line a' be assigned. Denote by h' a half-ray of the straight line a' emanating from a point O' of this line. Then in the plane α' there is one and only one half-ray k' such that the angle (h,k), or (k,h), is congruent to the angle (h',k') and at the same time all interior points of the angle (h',k') lie upon the given side of a'. We express this relation by means of the notation ∠ (h,k) ≡ ∠ (h',k') Every angle is congruent to itself; that is, ∠ (h,k) ≡ ∠ (h,k) or ∠ (h,k) ≡ ∠ (k,h) We say, briefly, that every angle in a given plane can be laid off upon a given side of a given half-ray in one and only one way.

en · unofficial · sha256:cb55100e7b71e2bd5d12c14c460826b9ec40c061a59567d6f346f9fe52ed0e50

IV, 5. If the angle (h,k) is congruent to the angle (h',k') and to the angle (h”,k”), then the angle (h',k') is congruent to the angle (h”,k”); that is to say, if ∠ (h, k) ≡ ∠ (h', k') and ∠ (h, k) ≡ ∠ (h”,k”), then ∠ (h',k') ≡ ∠ (h”,k”). Suppose we have given a triangle ABC. Denote by h, k the two half-rays emanating from A and passing respectively through B and C. The angle (h,k) is then said to be the angle included by the sides AB and AC, or the one opposite to the side BC in the triangle ABC. It contains all of the interior points of the triangle ABC and is represented by the symbol ∠ BAC, or by ∠ A.

en · unofficial · sha256:8660e8a9eeb7282625b53f9fe773a23be9e2b2a1aef018bbb22cf8feb184224d

IV, 6. If, in the two triangles ABC and A'B'C' the congruences AB ≡ A'B', AC ≡ A'C', ∠ BAC ≡ ∠ B'A'C' hold, then the congruences ∠ ABC ≡ ∠ A'B'C' and ∠ ACB ≡ ∠ A'C'B' also hold. Axioms IV, 1–3 contain statements concerning the congruence of segments of a straight line only. They may, therefore, be called the linear axioms of group IV. Axioms IV, 4, 5 contain statements relating to the congruence of angles. Axiom IV, 6 gives the connection between the congruence of segments and the congruence of angles. Axioms IV, 4–6 contain statements regarding the elements of plane geometry and may be called the plane axioms of group IV.

en · unofficial · sha256:a53106fa53a742c1c5e252d66b0e71f095b92a301997053dcef438873dd2fbc0

I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. Instead of “determine,” we may also employ other forms of expression; for example, we may say A “lies upon” a, A “is a point of” a, a “goes through” A “and through” B, a “joins” A “and” or “with” B, etc. If A lies upon a and at the same time upon another straight line b, we make use also of the expression: “The straight lines” a “and” b “have the point A in common,” etc.

Referencing norms: AxiomI1, Gerade, Punkt, VerbindendeGerade, liegt_auf, verbindende_gerade, verschiedene_punkte, HilbertReading/HilbertBaseChosen

en · unofficial · sha256:cd359d370c31cc64529df27cb3b9a954427d0351757702299663cfa8551e1965

I, 2. Any two distinct points of a straight line completely determine that line; that is, if AB = a and AC=a, where B ≠ C, then is also BC=a.

Referencing norms: AxiomI2, AxiomI2Eindeutig, geraden_fallen_zusammen, HilbertReading/HilbertBaseChosen

en · unofficial · sha256:3567a4f30918ffc9cc685632c78ccd43d08a42b52739915dcc172ab61c265e86

I, 3. Three points A, B, C not situated in the same straight line always completely determine a plane α. We write ABC=α. We employ also the expressions: A, B, C, “lie in” α; A, B, C “are points of” α, etc.

Referencing norms: AxiomI3, Ebene, Traegerebene, dreieck, liegt_in, traegerebene

en · unofficial · sha256:6cfb4fa017b2283b09efb753f0b1c611e785bc58f1ed03cc6deef8583e0defd6

I, 4. Any three points A, B, C of a plane α, which do not lie in the same straight line, completely determine that plane.

Referencing norms: AxiomI4, AxiomI4Eindeutig, ebenen_fallen_zusammen

en · unofficial · sha256:83bd822e09014cba8ef57792a044fb5d5abcd0f7cacbdb9ddb227a069f44373e

I, 5. If two points A, B of a straight line a lie in a plane α, then every point of a lies in a. In this case we say: “The straight line a lies in the plane α,” etc.

Referencing norms: AxiomI5, AxiomI5Ebenenzugehoerigkeit, LiegtInEbene, liegt_in_ebene

en · unofficial · sha256:6c3544b226d362fd8327afa6c1a1cf7c6b67c93b0367bc1b128cc416650b1d40

I, 6. If two planes α, β have a point A in common, then they have at least a second point B in common.

Referencing norms: AxiomI6, GemeinsamerPunkt, ZweiGemeinsamePunkte, gemeinsamer_punkt, zwei_gemeinsame_punkte

en · unofficial · sha256:abdf6c83a69c3adee474ea15f9a717e2dba8885d5f1ff8c911e403e28ebd1de6

I, 7. Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane. Axioms I, 1–2 contain statements concerning points and straight lines only; that is, concerning the elements of plane geometry. We will call them, therefore, the plane axioms of group I, in order to distinguish them from the axioms I, 3–7, which we will designate briefly as the space axioms of this group. Of the theorems which follow from the axioms I, 3–7, we shall mention only the following:

Referencing norms: AxiomI7Ebene, AxiomI7Gerade, AxiomI7Raum, DreiPunkteInEbene, Komplanar, VierPunkteAusserhalbEinerEbene, ZweiPunkteAuf, drei_punkte_in_ebene, komplanar, raeumliche_figur, vier_punkte_ausserhalb_einer_ebene, zwei_punkte_auf

en · unofficial · sha256:1d579f0d9f6729d840cab7858495029e190d4fe04ebe9210128b9eca902b809b

V. Let A₁ be any point upon a straight line between the arbitrarily chosen points A and B. Take the points A₂, A₃, A₄,… so that A₁ lies between A and A₂, A₂ between A₁ and A₃, A₃ between A₂ and A₄ etc. Moreover, let the segments A A₁, A₁ A₂, A₂ A₃, A₃ A₄, … be equal to one another. Then, among this series of points, there always exists a certain point Aₙ such that B lies between A and Aₙ. The axiom of Archimedes is a linear axiom.

en · unofficial · sha256:141cb7fd6a9132784ce2dd4992c698a7d11d4cb5e10745ff7569543707eac63e

Axiom of Completeness. Footnote — See Hilbert, “Ueber den Zahlenbegriff,” Berichte der deutschen Mathematiker-Vereinigung, 1900. (Vollständigkeit): To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid. This axiom gives us nothing directly concerning the existence of limiting points, or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano's theorem by virtue of which, for all sets of points situated upon a straight line between two definite points of the same line, there exists necessarily a point of condensation, that is to say, a limiting point. From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers. However, in what is to follow, no use will be made of the “axiom of completeness.” COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.