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Contents
I Classical Geometry 1
1 Absolute (Neutral) Geometry 3
1.1 Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Hilbert’s Axioms of Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Consequences of Incidence Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Betweenness and Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Hilbert’s Axioms of Betweenness and Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Basic Properties of Betweenness Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Betweenness Properties for n Collinear Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Every Open Interval Contains Infinitely Many Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Further Properties of Open Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Open Sets and Fundamental Topological Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Basic Properties of Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Linear Ordering on Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Ordering on Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Complementary Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Point Sets on Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Basic Properties of Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Point Sets on Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Complementary Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Basic Properties of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Definition and Basic Properties of Generalized Betweenness Relations . . . . . . . . . . . . . . . . . . 46
Further Properties of Generalized Betweenness Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Generalized Betweenness Relation for n Geometric Objects . . . . . . . . . . . . . . . . . . . . . . . . 53
Some Properties of Generalized Open Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Basic Properties of Generalized Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Linear Ordering on Generalized Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Linear Ordering on Sets With Generalized Betweenness Relation . . . . . . . . . . . . . . . . . . . . . 59
Complementary Generalized Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Sets of Geometric Objects on Generalized Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Betweenness Relation for Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Betweenness Relation For n Rays With Common Initial Point . . . . . . . . . . . . . . . . . . . . . . . 64
Basic Properties of Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Line Ordering on Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Line Ordering on Pencils of Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Complementary Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Sets of (Traditional) Rays on Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Paths and Polygons: Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Simplicity and Related Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Some Properties of Triangles and Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Basic Properties of Trapezoids and Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Basic Properties of Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Point Sets in Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Complementary Half-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Basic Properties of Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Betweenness Relation for Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Betweenness Relation for n Half-Planes with Common Edge . . . . . . . . . . . . . . . . . . . . . . . . 102
Basic Properties of Dihedral Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Linear Ordering on Dihedral Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Line Ordering on Pencils of Half-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Complementary Dihedral Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
iSets of Half-Planes on Dihedral Angular Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Properties of Convex Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
1.3 Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Hilbert’s Axioms of Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Basic Properties of Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Congruence of Triangles: SAS & ASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Congruence of Adjacent Supplementary and Vertical Angles . . . . . . . . . . . . . . . . . . . . . . . . 114
Right Angles and Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Congruence and Betweenness for Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Congruence and Betweenness for Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Congruence of Triangles:SSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Congruence of Angles and Congruence of Paths as Equivalence Relations . . . . . . . . . . . . . . . . 121
Comparison of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Generalized Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Comparison of Generalized Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Comparison of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Acute, Obtuse and Right Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Interior and Exterior Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Relations Between Intervals and Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
SAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Relations Between Intervals Divided into Congruent Parts . . . . . . . . . . . . . . . . . . . . . . . . . 144
Midpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Triangle Medians, Bisectors, and Altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Congruence and Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Right Bisectors of Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Isometries on the Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Isometries of Collinear Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
General Notion of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Comparison of Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Acute, Obtuse and Right Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
1.4 Continuity, Measurement, and Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Axioms of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
2 Elementary Euclidean Geometry 229
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
3 Elementary Hyperbolic (Lobachevskian) Geometry 235
3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4 Elementary Projective Geometry 249
Notation
Symbol Meaning
⇋ The symbol on the left of ⇋ equals by definition the expression on the right of ⇋.
def
⇐⇒ The expression on the left of
def
⇐⇒ equals by definition the expression on the right of
def
⇐⇒.
N The set of natural numbers (positive integers).
N
0
The set N
0 ⇋ {0} ∪ N of nonnegative integers.
Nn The set {1, 2, . . . , n}, where n ∈ N.
iiSymbol Meaning Page
A, B, C, . . . Capital Latin letters usually denote points. 3
a, b, c, . . . Small Latin letters usually denote lines. 3
α, β, γ, . . . Small Greek letters usually denote planes. 3
C
P t
The class of all points. 3
C
L
The class of all lines. 3
C
P l
The class of all planes. 3
aAB Line drawn through A, B. 3
αABC Plane incident with the non-collinear points A, B, C 3
Pa ⇋ {A|A ∈ a} The set of all points (”contour”) of the line a 3
Pα ⇋ {A|A ∈ α} The set of all points (”contour”) of the plane α 3
a ⊂ α Line a lies on plane α, plane α goes through line a. 3
X ⊂ Pa The figure (geometric object) X lies on line a. 3
X ⊂ Pα The figure (geometric object) X lies on plane α. 3
A ∈ a ∩ b Line a meets line b in a point A 4
A ∈ a ∩ β Line a meets plane β in a point A. 4
A ∈ a ∩ B Line a meets figure B in a point A. 4
A ∈ a ∩ B Figure A meets figure B in a point A. 4
αaA Plane drawn through line a and point A. 5
a k b line a is parallel to line b, i.e. a, b coplane and do not meet. 6
ab an abstract strip ab is a pair of parallel lines a, b. 6
a k α line a is parallel to plane α, i.e. a, α do not meet. 6
α k β plane α is parallel to plane β, i.e. α, β do not meet. 6
αab Plane containing lines a, b, whether parallel or having a common point. 7
[ABC] Point B lies between points A, C. 7
AB (Abstract) interval with ends A, B, i.e. the set {A, B}. 7
(AB) Open interval with ends A, B, i.e. the set {C|[ACB]}. 7
[AB) Half-open interval with ends A, B, i.e. the set (AB) ∪ {A, B}. 7
(AB] Half-closed interval with ends A, B, i.e. the set (AB) ∪ {B}. 7
[AB] Closed interval with ends A, B, i.e. the set (AB) ∪ {A, B}. 7
IntX Interior of the figure (point set) X. 7
ExtX Exterior of the figure (point set) X. 7
[A1A2 . . . An . . .] Points A1, A2, . . . , An, . . ., where n ∈ N, n ≥ 3 are in order [A1A2 . . . An . . .]. 15
OA Ray through O emanating from A, i.e. OA ⇋ {B|B ∈ aOA & B =6 O & ¬[AOB]}. 18
h¯ The line containing the ray h. 18
O = ∂h The initial point of the ray h. 18
(A ≺ B)OD , A ≺ B Point A precedes the point B on the ray OD, i.e. (A ≺ B)OD
def
⇐⇒ [OAB]. 21
A B A either precedes B or coincides with it, i.e. A B
def
⇐⇒ (A ≺ B) ∨ (A = B). 21
(A ≺ B)a, A ≺ B Point A precedes point B on line a. 22
(A≺1B)a A precedes B in direct order on line a. 22
(A≺2B)a A precedes B in inverse order on line a. 22
Oc
A Ray, complementary to the ray OA. 25
(ABa)α, ABa Points A, B lie (in plane α) on the same side of the line a. 27
(AaB)α, AaB Points A, B lie (in plane α) on opposite sides of the line a. 27
aA Half-plane with the edge a and containing the point A. 27
(ABa)α, ABa Point sets (figures) A, B lie (in plane α) on the same side of the line a. 29
(AaB)α, AaB Point sets (figures) A, B lie (in plane α) on opposite sides of the line a. 29
aA Half-plane with the edge a and containing the figure A. 29
a
c
A Half-plane, complementary to the half-plane aA. 30
χ¯ the plane containing the half-plane χ. 32
∠(h, k)O, ∠(h, k) Angle with vertex O (usually written simply as ∠(h, k)). 35
P∠(h,k) Set of points, or contour, of the angle ∠(h, k)O, i.e. the set h ∪ {O} ∪ k. 36
Int∠(h, k) Interior of the angle ∠(h, k). 36
adj∠(h, k) Any angle, adjacent to ∠(h, k). 38
adjsp ∠(h, k) Any of the two angles, adjacent supplementary to the angle .∠(h, k) 39
vert ∠(h, k) Angle ∠(h
c
, k
c
), vertical to the angle ∠(h, k). 40
[ABC] Geometric object B lies between geometric objects A, C. 46
AB Generalized (abstract) interval with ends A, B, i.e. the set {A, B}. 48
(AB) Generalized open interval with ends A, B, i.e. the set {C|[ACB]}. 48
[AB) Generalized half-open interval with ends A, B, i.e. the set (AB) ∪ {A, B}. 48
(AB] Generalized half-closed interval with ends A, B, i.e. the set (AB) ∪ {B}. 48
[AB]. Generalized closed interval with ends A, B, i.e. the set (AB) ∪ {A, B}. 48
P
(O) A ray pencil, i.e. a collection of rays emanating from the point O. 48
iiiSymbol Meaning Page
[hkl] Ray k lies between rays h, l. 48
∠(h, h
c
) A straight angle (with sides h, h
c
). 49
[A1A2 . . . An(. . .)] Geometric objects A1, A2, . . . , An(, . . .) are in order [A1A2 . . . An(. . .)] 54
O
(J)
A , OA Generalized ray drawn from O through A. 56
(A ≺ B)OD The geometric object A precedes the geometric object B on OD. 58
A B For A, B on OD we let A B
def
⇐⇒ (A ≺ B) ∨ (A = B) 58
(A≺iB)J A precedes B in J in the direct (i = 1) or inverse (i = 2) order. 59
AiB For A, B in J we let AiB
def
⇐⇒ (A≺iB) ∨ (A = B) 60
O
c(J)
A , Oc
A The generalized ray Oc
A, complementary in J to the generalized ray OA. 61
(hk) Open angular interval. 63
[hk). Half-open angular interval. 63
(hk] Half-closed angular interval. 63
[hk] Closed angular interval. 63
[h1h2 . . . hn(. . .)] The rays h1, h2, . . . , hn(, . . .) are in order [h1h2 . . . hn(. . .)]. 65
oh Angular ray emanating from the ray o and containing the ray h 65
(h ≺ k)om, h ≺ k The ray h precedes the ray k on the angular ray om. 66
(h k)om, h k For rays h, k on an angular ray om we let h k
def
⇐⇒ (h ≺ k) ∨ (h = k) 66
(h≺ik) The ray h precedes the ray k in the direct (i = 1) or inverse (i = 2)order. 66
o
c
h The ray, complementary to the angular ray oh. 68
−−→
AB An ordered interval. 24
A0A1 . . . An A (rectilinear) path A0A1 . . . An. 69
A0A1 . . . An A polygon, i.e. the (rectilinear) path A0A1 . . . AnAn+1 with An+1 = A0. 69
△ABC A triangle with the vertices A, B, C. 69
(A ≺ B)A1A2...An
, A ≺ B A precedes B on the path A1A2 . . . An. 70
∠Ai−1AiAi+1, ∠Ai Angle between sides Ai−1Ai
, AiAi+1 of the path/polygon A0A1 . . . AnAn+1. 71
ABα Points A, B lie on the same side of the plane α. 80
AαB Points A, B lie on opposite sides of the plane α. 80
αA Half-space, containing the point A, i.e. αA ⇋ {B|ABα}. 80
ABα Figures (point sets) A, B lie on the same side of the plane α. 81
AαB Figures (point sets) A, B lie on opposite sides of the plane α. 81
α
c
A Half-space, complementary to the half-space αA. 82
(χκc)a, χκc A dihedral formed by the half-planes χ, κ with the common edge a. 87
P(χκc ) The set of points of the dihedral angle (χκc)a, i.e. P(χκc ) ⇋ χ ∪ Pa ∪ κ. 88
adj(χκc) Any dihedral angle, adjacent to the given dihedral angle χκc 91
adjsp χκc Any of the two dihedral angles, adjacent supplementary to χκc. 91
vert(χκc) The dihedral angle, vertical to χκc, i.e. vert(χκc) ⇋ χ[c
κ
c
. 94
S
(a) A pencil of half-planes with the same edge a. 98
[aAaBaC] Half-plane aB lies between the half-planes aA, aC. 98
(aAaC) Open dihedral angular interval formed by the half-planes aA, aC. 99
[aAaC) Half-open dihedral angular interval formed by the half-planes aA, aC. 99
(aAaC] Half-closed dihedral angular interval formed by the half-planes aA, aC. 99
[aAaC] Closed dihedral angular interval formed by the half-planes aA, aC. 99
[χ1χ2 . . . χn(. . .)] The half-planes χ1, χ2, . . . , χn(, . . .) are in order [χ1χ2 . . . χn(. . .)]. 104
oχ Dihedral angular ray emanating from o and containing χ. 104
(χ ≺ κ)oµ The half-plane χ precedes the half-plane κ on the dihedral angular ray oµ. 105
(χ κ)oµ
, χ κ For half-planes χ, κ on oµ we let χ κ
def
⇐⇒ (χ ≺ κ) ∨ (χ = κ). 106
(χ≺iκ) The half-plane χ precedes κ in the direct (i = 1) or inverse (i = 2)order. 106
o
c
χ Dihedral angular ray, complementary to the dihedral angular ray oχ. 107
AB ≡ CD The interval AB is congruent to the interval CD 110
∠(h, k) ≡ ∠(l, m) Angle ∠(h, k) is congruent to the angle ∠(l, m) 110
A ≡ B The figure (point set) A is congruent to the figure B. 110
A1A2 . . . An ≃ B1B2 . . . Bn The path A1A2 . . . An is weakly congruent to the path B1B2 . . . Bn. 110
A1A2 . . . An ≡ B1B2 . . . Bn The path A1A2 . . . An is congruent to the path B1B2 . . . Bn. 110
A1A2 . . . An =∼ B1B2 . . . Bn The path A1A2 . . . An is strongly congruent to the path B1B2 . . . Bn. 111
a ⊥ b The line a is perpendicular to the line b. 117
proj(A, a) Projection of the point A on the line a. 117
proj(AB, a) Projection of the interval AB on the line a. 117
AB ≧ A′B′
The interval A′B′
is shorter than or congruent to the interval AB. 124
A′B′ < AB The interval A′B′
is shorter than the interval AB. 124
∠(h
′
, k
′
) ≦ ∠(h, k) The angle ∠(h
′
, k
′
) is less than or congruent to the angle ∠(h, k). 126
∠(h
′
, k
′
) < ∠(h, k) The angle ∠(h
′
, k
′
) is less than the angle ∠(h, k). 126
ivSymbol Meaning Page
AB ≡ CD The generalized interval AB is congruent to the generalized interval CD. 126
AB ≧ A′
B
′
The generalized interval AB is shorter than or congruent to the generalized interval A′
B
′
. 129
AB < A′
B
′
The generalized interval AB is shorter than the generalized interval A′
B
′
. 129
E = mid AB The point E is the midpoint of the interval AB.
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