Title: Theory / Unimportant / Math / Geometry/ / Non-Euclidean - Coolidge;
Tags: fromImplication counterExample complex fromMath

Question: The undefined or geometry are
Answer: A set of points \(P\),
A set of distances \(D\)
A function \(\overline{\cdot,\cdot}:P\times P\to D\)
A congruence relation \(\equiv\) on \(D\),
A triadic relations on distances \(\cdot\equiv\cdot+\cdot\)
Page: 13
Chapter: 1

Definition: Points
Page: 14
Chapter: 1
Index: I
Kind: Axiom
Denoted by: Large roman letters, \(A, B, C\)

Name: The distance between \(A\) and \(B\)
Page: 14
Chapter: 1
Variables: \(A,B\) points
Notation: \(\overline{AB}\)
Name2: The distance from \(A\) to \(B\)

Name: The distance \(\overline{AB}\) and \(\overline{CD}\) belongs to the congruence relation
Page: 14
Chapter: 1
Variables: \(S\) a set of points,
\(A,B,C,D\) points
Notation: \(\overline{AB}\equiv\overline{CD}\)

Name: The set of distances
Typ: The non-negative part of a dense subgroup \(S\) of \((\mathbb R_{\ge0},+)\), without maximal element
Page: none
Chapter: 1
Index: III to XI
Kind: Axiom
Extra: Not in the book, but inferred from the context

Name: The segment of \(A\) and \(B\)
Definition: The set of all points \(C\) possessing the property that \(\overline{AB}\equiv\overline{AC}+\overline{CB}\)
Page: none
Chapter: 1
Variables: \(A,B\) points
Notation: \((AB)\)

Name: The extremities of \((AB)\)
Definition: \(A\) and \(B\)
Page: 15
Chapter: 1
Variables: \(A,B\) points,

Name: The points within \((AB)\)
Definition: The segment \((AB)\) without its extremities point
Page: 15
Chapter: 1
Variables: \(A,B\) points,

Name: The extension of \((AB)\) beyond \(B\)
Definition: The set of points \(C\) such that \(\overline{AC}=\overline{AB}+\overline{BC}\)
Page: 15
Chapter: 1
Variables: \(A,B\) points,

Smaller0: \(C,D\in (AB)\)
Greater0: \((CD)\subseteq (AB)\)
Page: 16
Chapter: 1
Smaller to greater: iff

Smaller0: \(C\in (AB)\)
Greater0: \(\{(AC)\setminus\{C\},(CB)\setminus\{C\}\}\)
Page: 16
Chapter: 1
Index: 4
Kind: Theo
Greater1: a partition of \((AB)\setminus\{C\}\)
Connect Greater0: is

Name: A line
Definition: The union of a segment and its extension
Page: 16
Chapter: 1
Index: 4
Kind: Def

Name: Two lines which intersects
Definition: Two lines having in common a single point
Page: 16
Chapter: 1
Kind: Def
Name2: Two lines which cuts

Definition: The lines determined by the points \(A\) and \(B\)
Page: 17
Chapter: 1
Context: Geometry
Notation: \(AB\)
Kind: Def

Smaller0: \(C,D\) are points of the extension of \((AB)\) beyond \(B\)
Greater0: \(D\in(BC)\)
Page: 17
Chapter: 1
Index: 10
Kind: Theo
Smaller1: \(\overline{BC}\ge\overline{BD}\)
Greater1: \(D\) is an extension of \(BC\)

Question: There exists 3 points which
Answer: are not on the same line.
Page: 17
Chapter: 1
Index: XV
Kind: Axiom

Question: The number of points is
Answer: at least 2
Page: 14
Chapter: 1
Index: I
Kind: Axiom

Question: \(\overline{AA}\)
Answer: equals 0
Page: 14
Chapter: 1
Index: IX
Kind: Axiom

Smaller0: \(\overline{AB}\equiv\overline{PQ}+\overline{RS}\)
Greater0: there is a single point \(C\) of \((AB)\) such that \(\overline{AC}\equiv\overline{PQ}\) and \(\overline{CB}\equiv \overline{RS}\)
Page: 15
Chapter: 1
Variables: \(A,B,P,Q,R,S\) points
Index: XII
Kind: Axiom
Hide middle: yes

Smaller0: at least 2
Greater0: \(L=L'\)
Page: 17
Chapter: 1
Variables: \(L,L'\) two lines
Index: XIV
Kind: Axiom
Smaller to greater: implies
Extra: Reverse seems false if a line has a single point
Prefix Smaller: \(L\cap L'\) cardinal is

Smaller0: \(B\in (CD)\)
Greater0: \((AC)\).
Page: 17
Chapter: 1
Variables: \(A,B,C,D,E\) points
Index: XVI
Kind: Axiom
Prefix Greater: \(DE\) intersects
Smaller1: \(A\not\in BC\)
Smaller2: \(E\in (AB)\)

Question: The number of point on \((A,B)\) is
Answer: infinite
Page: 18
Chapter: 1
Variables: \(A,B\) points
Index: 11
Kind: Theo

Name: A set of collinear point
Definition: A set of points on the same line
Page: 18
Chapter: 1
Kind: Def

Name: A set of concurrent lines
Definition: A set of lines passing through the same point
Page: 18
Chapter: 1
Kind: Def

Name: A line passing through \(A\)
Definition: A line \(L\) such that \(A\in L\)
Page: 18
Chapter: 1
Variables: \(A\) a point
Kind: Def

Smaller0: \(B_i\in (A_i,C)\)
Greater0: the \((B_iA_{3-i})\) intersects
Page: 18
Chapter: 1
Variables: \(A_1,B_1,C\) and \(A_2,B_2,C\) triplets of aligned points
Index: 14
Kind: Theo