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Euclidean Geometry and its subgeometries - Coggle Diagram
Euclidean Geometry and its subgeometries
Incidence Geometry
Definitions
Definition I.0.
Space U is the set of all points. We may think of space as the
universe or the universe of discourse.
Definition I.0.1.
(A) “Points A, B, and C are
collinear
means that there is a line L such that A, B, and C all lie on line L. More generally, if E is any set of points, then E is
collinear
iff there exists a line L such that E \(\subseteq\) L. A set E is
noncollinear
iff there is no line containing all the points of E.
(B) “Points A, B, C, and D are
coplanar
means that there is a plane P such that A, B, C, and D all lie on P. More generally, if E is any set of points, then E is coplanar iff there exists a plane P such that E \(\subseteq\) P . A set E is
noncoplanar
iff there is no plane containing all the points of E.
(C) If E is a set of two or more lines, the lines in E are said to be
concurrent at a point
O if and only if the intersection of all members of E is {O}.
(D) A space on which the
incidence axioms
I.0 through I.5 are true is an incidence space, and a plane therein is an incidence plane. The geometry these axioms generate is incidence geometry.
Theorems
Theorem I.3.
If E and F are distinct planes both of which contain line L, then E \( \cap \) F = L.
proof page 28
Theorem I.4.
If the intersection of two distinct planes is nonempty, then it is a line.
proof page 29
Theorem I.5.
Given a plane E and a point A belonging to E, there exists a line L such that L \(\subseteq\) E and A \( \not\in \) L.
proof page 29
Theorem I.6
(Two intersecting lines determine a plane).
Given lines L and M such that L \( \neq \) M and L \( \cap \) M \( \neq \emptyset \) there exists one and only one plane E such that L \(\subseteq\) E and M \(\subseteq\) E .
proof page 29
Theorem I.7.
Let E be a plane. There exists a point P such that P \( \not\in \) E .
proof page 30
Theorem I.8.
Let
S
and
T
be distinct planes whose
intersection
is the line
L
, and let
P
be a member of
L
.
Then there exist lines M and N such that M \(\subseteq\) S, N \(\subseteq\) T, M \( \neq\) L , N \( \neq\) L , and M \( \cap\) N = {P}.
If M and N are any two lines satisfying these conditions, then there is exactly one plane E such that M \( \cup\) N \(\subseteq\) E . Moreover, E \( \cap\) L = {P}
proof page 30
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Axioms
Axiom I.1.
There exists exactly one line through two distinct points.
Axiom I.2.
There exists exactly one plane through three noncollinear points.
Axiom I.3.
If two distinct points lie in a plane, then any line through the points is contained in the plane.
Axiom I.4.
If two distinct planes have a nonempty intersection, then their intersection has at least two members.
Axiom I.5.
(A) There exist at least two distinct points on every line.
(B) There exists at least one noncollinear set of three points on every plane.
(C) There exists at least one noncoplanar set of four points in space.
Affine Geometry: Incidence with Parallelism (IP)
Definitions
Definition IP.0.
(A) Lines L and M are
parallel
(notation: \(L \parallel M \) ) iff there is a plane that contains them both and \(L \cap M = \emptyset \).
(B) A line L and a plane P are
parallel
(notation: \(L \parallel P) \) iff \(L \cap P = \emptyset \)
(C) Two planes P and P 0 are
parallel
(notation: \(P \parallel P') \))iff \(P \cap P' = \emptyset \)
(D) A set E of two or more distinct lines on a plane P is a
pencil
iff either
(1) the members of E are concurrent at some point O, or
(2) every member of E is parallel to every other member of E. In case (1), the point O is the focal point of E.
definitions on page 37-38
Definition IP.1.
A plane P is an
affine plane
iff it is a subset of space where the
incidence axioms
and
Axiom PS
hold.
Affine geometry
is the term used to describe the geometry of such a plane.
definitions on page 40
Theorems
Theorem IP.2.
Let E be a plane, and let M and L be parallel lines; if L \(\subseteq\) E and M \(\not\subseteq\) E , then M \( \parallel \) E .
proof page 40
Theorem IP.3.
Let L be a line in a plane F and suppose L is
parallel
to a plane E that intersects F . Then E \(\cap\) F is a line M which is parallel to L .
proof page 40
Theorem IP.4.
If F is a plane containing two lines L and M which intersect at a point P, and if E is a plane which is parallel to both L and M , then E \( \parallel \) F .
proof page 41
Theorem IP.5.
Given a plane E and a point P not on E , there exists exactly one plane F such that P 2 F and E \( \parallel \) F .
proof page 41
Theorem IP.6
(
Transitivity of Parallelism
). If L , M , and N are distinct lines such that L \( \parallel\) M and M \( \parallel\) N , then L \( \parallel\) N .
proof page 42
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Axioms
incidence axioms
Axiom PS (Strong Form of the Parallel Axiom)
Given a line L and a point P not belonging to L, there exists exactly one line M such that \(P \in M \) and \(L\parallel M \) . (If such a line exists, it is denoted as \( par(L,M) \) )
Axiom PW (Weak Form of the Parallel Axiom).
Given a line L and a point P not belonging to L , there exists at most one line M such that \(P \in M \) and \(L\parallel M \) . Note that Axiom PW does not guarantee that such a line exists.
Collineations of an Affine Plane (CAP)
Definitions
definitions on page 46
definitions on page 49
definitions on page 50
definitions on page 54
definitions on page 58
Theorems
proof on page 47
proof on page 48
proof on page 48
proof on page 48
proof on page 49
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Incidence and Betweenness Space
This chapter defines a betweenness relation and uses it to define segments, rays, and triangles.
A few theorems are proved in the resulting IB geometry. These are foundational for the rest of the development.
Definitions
definition on page 64
definition on page 64
definition on page 66
definition on page 67
definition on page 67
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Axioms
Axiom Bet:
There exists a betweenness relation.
Theorems
proof on page 69
proof on page 70
proof on page 71
proof on page 73
proof on page 74
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Pasch Geometry
Definitions
definition on page 86
Axioms
Theorems
proof page 83
proof page 83
proof page 83
proof page 84
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