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Fundamentals of GeometryOleg A. Belyaevbelyaev@polly.phys.msu.ruFebruary 28, 2007

ContentsIClassical Geometry11 Absolute (Neutral) Geometry1.1 Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Hilbert’s Axioms of Incidence . . . . . . . . . . . . . . . . . . . . . . .Consequences of Incidence Axioms . . . . . . . . . . . . . . . . . . . .1.2 Betweenness and Order . . . . . . . . . . . . . . . . . . . . . . . . . .Hilbert’s Axioms of Betweenness and Order . . . . . . . . . . . . . . .Basic Properties of Betweenness Relation . . . . . . . . . . . . . . . .Betweenness Properties for n Collinear Points . . . . . . . . . . . . . .Every Open Interval Contains Infinitely Many Points . . . . . . . . . .Further Properties of Open Intervals . . . . . . . . . . . . . . . . . . .Open Sets and Fundamental Topological Properties . . . . . . . . . . .Basic Properties of Rays . . . . . . . . . . . . . . . . . . . . . . . . . .Linear Ordering on Rays . . . . . . . . . . . . . . . . . . . . . . . . . .Ordering on Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Complementary Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . .Point Sets on Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Basic Properties of Half-Planes . . . . . . . . . . . . . . . . . . . . . .Point Sets on Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . .Complementary Half-Planes . . . . . . . . . . . . . . . . . . . . . . . .Basic Properties of Angles . . . . . . . . . . . . . . . . . . . . . . . . .Definition and Basic Properties of Generalized Betweenness RelationsFurther Properties of Generalized Betweenness Relations . . . . . . . .Generalized Betweenness Relation for n Geometric Objects . . . . . .Some Properties of Generalized Open Intervals . . . . . . . . . . . . .Basic Properties of Generalized Rays . . . . . . . . . . . . . . . . . . .Linear Ordering on Generalized Rays . . . . . . . . . . . . . . . . . . .Linear Ordering on Sets With Generalized Betweenness Relation . . .Complementary Generalized Rays . . . . . . . . . . . . . . . . . . . . .Sets of Geometric Objects on Generalized Rays . . . . . . . . . . . . .Betweenness Relation for Rays . . . . . . . . . . . . . . . . . . . . . .Betweenness Relation For n Rays With Common Initial Point . . . . .Basic Properties of Angular Rays . . . . . . . . . . . . . . . . . . . . .Line Ordering on Angular Rays . . . . . . . . . . . . . . . . . . . . . .Line Ordering on Pencils of Rays . . . . . . . . . . . . . . . . . . . . .Complementary Angular Rays . . . . . . . . . . . . . . . . . . . . . . .Sets of (Traditional) Rays on Angular Rays . . . . . . . . . . . . . . .Paths and Polygons: Basic Concepts . . . . . . . . . . . . . . . . . . .Simplicity and Related Properties . . . . . . . . . . . . . . . . . . . . .Some Properties of Triangles and Quadrilaterals . . . . . . . . . . . .Basic Properties of Trapezoids and Parallelograms . . . . . . . . . . .Basic Properties of Half-Spaces . . . . . . . . . . . . . . . . . . . . . .Point Sets in Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . .Complementary Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . .Basic Properties of Dihedral Angles . . . . . . . . . . . . . . . . . . .Betweenness Relation for Half-Planes . . . . . . . . . . . . . . . . . . .Betweenness Relation for n Half-Planes with Common Edge . . . . . .Basic Properties of Dihedral Angular Rays . . . . . . . . . . . . . . . .Linear Ordering on Dihedral Angular Rays . . . . . . . . . . . . . . .Line Ordering on Pencils of Half-Planes . . . . . . . . . . . . . . . . .Complementary Dihedral Angular Rays . . . . . . . . . . . . . . . . 7

Sets of Half-Planes on Dihedral Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Properties of Convex Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081.3Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Hilbert’s Axioms of Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Basic Properties of Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Congruence of Triangles: SAS & ASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Congruence of Adjacent Supplementary and Vertical Angles . . . . . . . . . . . . . . . . . . . . . . . . 114Right Angles and Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Congruence and Betweenness for Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Congruence and Betweenness for Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Congruence of Triangles:SSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Congruence of Angles and Congruence of Paths as Equivalence Relations . . . . . . . . . . . . . . . . 121Comparison of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Generalized Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Comparison of Generalized Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Comparison of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Acute, Obtuse and Right Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Interior and Exterior Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Relations Between Intervals and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136SAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Relations Between Intervals Divided into Congruent Parts . . . . . . . . . . . . . . . . . . . . . . . . . 144Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Triangle Medians, Bisectors, and Altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Congruence and Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Right Bisectors of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Isometries on the Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Isometries of Collinear Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166General Notion of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Comparison of Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Acute, Obtuse and Right Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1891.4Continuity, Measurement, and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Axioms of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2082 Elementary Euclidean Geometry2.1229. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293 Elementary Hyperbolic (Lobachevskian) Geometry3.1235. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2354 Elementary Projective Geometry249NotationSymbol def NN0NnMeaningThe symbol on the left of equals by definition the expression on the right of .TheTheTheThedefdefexpression on the left of equals by definition the expression on the right of .set of natural numbers (positive integers).set N0 {0} N of nonnegative integers.set {1, 2, . . . , n}, where n N.ii

SymbolA, B, C, . . .a, b, c, . . .α, β, γ, . . .CP tCLCP laABαABCPa {A A a}Pα {A A α}a αX PaX PαA a bA a βA a BA a ExtX[A1 A2 . . . An . . .]OAh̄O hMeaningCapital Latin letters usually denote points.Small Latin letters usually denote lines.Small Greek letters usually denote planes.The class of all points.The class of all lines.The class of all planes.Line drawn through A, B.Plane incident with the non-collinear points A, B, CThe set of all points (”contour”) of the line aThe set of all points (”contour”) of the plane αLine a lies on plane α, plane α goes through line a.The figure (geometric object) X lies on line a.The figure (geometric object) X lies on plane α.Line a meets line b in a point ALine a meets plane β in a point A.Line a meets figure B in a point A.Figure A meets figure B in a point A.Plane drawn through line a and point A.line a is parallel to line b, i.e. a, b coplane and do not meet.an abstract strip ab is a pair of parallel lines a, b.line a is parallel to plane α, i.e. a, α do not meet.plane α is parallel to plane β, i.e. α, β do not meet.Plane containing lines a, b, whether parallel or having a common point.Point B lies between points A, C.(Abstract) interval with ends A, B, i.e. the set {A, B}.Open interval with ends A, B, i.e. the set {C [ACB]}.Half-open interval with ends A, B, i.e. the set (AB) {A, B}.Half-closed interval with ends A, B, i.e. the set (AB) {B}.Closed interval with ends A, B, i.e. the set (AB) {A, B}.Interior of the figure (point set) X .Exterior of the figure (point set) X .Points A1 , A2 , . . . , An , . . ., where n N, n 3 are in order [A1 A2 . . . An . . .].Ray through O emanating from A, i.e. OA {B B aOA & B 6 O & [AOB]}.The line containing the ray h.The initial point of the ray h.(A B)OD , A BPoint A precedes the point B on the ray OD , i.e. (A B)OD [OAB].A B(A B)a , A B(A 1 B)a(A 2 B)acOA(ABa)α , ABa(AaB)α , AaBaA(ABa)α , ABa(AaB)α , AaBaAacAχ̄ (h, k)O , (h, k)P (h,k)Int (h, k)adj (h, k)adjsp (h, k)vert (h, k)[ABC]AB(AB)[AB)(AB][AB].P (O)A either precedes B or coincides with it, i.e. A B (A B) (A B).Point A precedes point B on line a.A precedes B in direct order on line a.A precedes B in inverse order on line a.Ray, complementary to the ray OA .Points A, B lie (in plane α) on the same side of the line a.Points A, B lie (in plane α) on opposite sides of the line a.Half-plane with the edge a and containing the point A.Point sets (figures) A, B lie (in plane α) on the same side of the line a.Point sets (figures) A, B lie (in plane α) on opposite sides of the line a.Half-plane with the edge a and containing the figure A.Half-plane, complementary to the half-plane aA .the plane containing the half-plane χ.Angle with vertex O (usually written simply as (h, k)).Set of points, or contour, of the angle (h, k)O , i.e. the set h {O} k.Interior of the angle (h, k).Any angle, adjacent to (h, k).Any of the two angles, adjacent supplementary to the angle . (h, k)Angle (hc , k c ), vertical to the angle (h, k).Geometric object B lies between geometric objects A, C.Generalized (abstract) interval with ends A, B, i.e. the set {A, B}.Generalized open interval with ends A, B, i.e. the set {C [ACB]}.Generalized half-open interval with ends A, B, i.e. the set (AB) {A, B}.Generalized half-closed interval with ends A, B, i.e. the set (AB) {B}.Generalized closed interval with ends A, B, i.e. the set (AB) {A, B}.A ray pencil, i.e. a collection of rays emanating from the point 48484848