Incidence

• Axioms:
  • $I_1$, Foreach line, $r$, exists a point on the line - and a point off the line: \[ \forall r \in \mathcal{R}: \qquad \exists P \in \mathcal{P} \quad\wedge\quad \exists P \notin \mathcal{P} \]
  • $I_2$, two points defines a line (uniquely): \[ \forall P \neq Q \in \mathcal{P}: \qquad \exists ! r \in \mathcal{R}: \qquad P, Q \in r \]
  • $I_3$: Exists two (distinct) Points. \[ \exists P,Q \in \mathcal{P}: ~p \neq Q \]
• Theorems:
  • $\mathcal{P} \neq \emptyset$
  • $ \exists P,Q,R \in \mathcal{P}$ distintos.
  • $ \exists r,s,t \in \mathcal{R}$ distintos.
• Models:
  • 3 Points and 3 Lines.
  • 4 Points and 6 Lines.

TiKZ Listing: Fig.tikz.tex. PDF   PNG   SVG   ZIP*   \tikzmath{\t=-0.15;}; \tikzmath{\tt=-0.25;}; \coordinate (A) at (0,0); \coordinate (B) at (2,0); \coordinate (C) at (0,2); \coordinate (D) at (2.0,2.0); \coordinate (V) at (4,0); %Draw 3 Nodes \filldraw (A) circle(1pt); \filldraw (B) circle(1pt); \filldraw (C) circle(1pt); %Node Titles \draw ($(A)!\tt!(C)$) node {$A$}; \draw ($(B)!\tt!(C)$) node {$B$}; \draw ($(C)!\tt!(B)$) node {$C$}; %Draw 3 Lines \draw ($(A)!\t!(B)$) -- ($(B)!\t!(A)$); \draw ($(A)!\t!(C)$) -- ($(C)!\t!(A)$); \draw ($(C)!\t!(B)$) -- ($(B)!\t!(C)$); \coordinate (A) at ($(A)+(V)$); \coordinate (B) at ($(B)+(V)$); \coordinate (C) at ($(C)+(V)$); \coordinate (D) at ($(D)+(V)$); %Draw 4 Nodes \filldraw (A) circle(1pt); \filldraw (B) circle(1pt); \filldraw (C) circle(1pt); \filldraw (D) circle(1pt); %Node Titles \draw ($(A)!\tt!(C)$) node {$A$}; \draw ($(B)!\tt!(C)$) node {$B$}; \draw ($(C)!\tt!(B)$) node {$C$}; \draw ($(D)!\tt!(C)$) node {$D$}; %Draw 6 Lines \draw ($(A)!\t!(B)$) -- ($(B)!\t!(A)$); \draw ($(A)!\t!(C)$) -- ($(C)!\t!(A)$); \draw ($(C)!\t!(B)$) -- ($(B)!\t!(C)$); \draw ($(D)!\t!(A)$) -- ($(A)!\t!(D)$); \draw ($(D)!\t!(B)$) -- ($(B)!\t!(D)$); \draw ($(D)!\t!(C)$) -- ($(C)!\t!(D)$);

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