Oleg A. Belyaev - Fundamentals of Geometry.pdf

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Fundamentals of Geometry
Fundamentals of Geometry
Oleg A. Belyaev
belyaev@polly.phys.msu.ru
February 28, 2007
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
i
Sets 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.
N n
The set{1, 2, . . . , n}, where n∈N.
ii
Symbol
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
a AB
Line drawn through A, B.
3
α ABC
Plane incident with the non-collinear points A, B, C
3
P a ⇋{A|A∈a} The set of all points (”contour”) of the line a
3
P ⇋{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⊂P a
The figure (geometric object)Xlies on line a.
3
X⊂P
The figure (geometric object)Xlies 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 figureBin a point A.
4
A∈a∩B FigureAmeets figureBin a point A.
4
α aA
Plane drawn through line a and point A.
5
ab
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
aα
line a is parallel to plane α, i.e. a, α do not meet.
6
αβ
plane α is parallel to plane β, i.e. α, β do not meet.
6
α ab
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
[A 1 A 2 . . . A n . . .]
Points A 1 , A 2 , . . . , A n , . . ., where n∈N, n≥3 are in order [A 1 A 2 . . . A n . . .].
15
O A
Ray through O emanating from A, i.e. O A ⇋{B|B∈a OA & B = O &¬[AOB]}.
18
The line containing the ray h.
18
O = ∂h
The initial point of the ray h.
18
(A≺B) O D , A≺B Point A precedes the point B on the ray O D , i.e. (A≺B) O D
def
⇐⇒[OAB].
21
def
⇐⇒(A≺B)∨(A = B).
AB
A either precedes B or coincides with it, i.e. AB
21
(A≺B) a , A≺B
Point A precedes point B on line a.
22
(A≺ 1 B) a
A precedes B in direct order on line a.
22
(A≺ 2 B) a
A precedes B in inverse order on line a.
22
O c A
Ray, complementary to the ray O A .
25
(ABa) , ABa
Points A, B lie (in plane α) on the same side of the line a.
27
(AaB) , AaB
Points A, B lie (in plane α) on opposite sides of the line a.
27
a A
Half-plane with the edge a and containing the point A.
27
(ABa) ,ABa
Point sets (figures)A,Blie (in plane α) on the same side of the line a.
29
(AaB) ,AaB Point sets (figures)A,Blie (in plane α) on opposite sides of the line a.
29
a A Half-plane with the edge a and containing the figureA.
29
a c A
Half-plane, complementary to the half-plane a A .
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 objectBlies between geometric objectsA,C.
46
AB Generalized (abstract) interval with endsA,B, i.e. the set{A,B}.
48
(AB)
Generalized open interval with endsA,B, i.e. the set{C|[ACB]}.
48
[AB)
Generalized half-open interval with endsA,B, i.e. the set (AB)∪{A,B}.
48
(AB]
Generalized half-closed interval with endsA,B, i.e. the set (AB)∪{B}.
48
[AB].
Generalized closed interval with endsA,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
iii
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