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\documentclass[]{scrartcl}

\usepackage{tikz}
\usetikzlibrary{calc}
\newcommand\Punkt{\tikz[scale=0.07]\draw[thick](-1,-1)--(1,1)(-1,1)--(1,-1);}


\begin{document}
	\begin{tikzpicture}
		\draw (0,0)--(3.9,0);
			\node at (2.2,1.5) (P) {\Punkt};
			\node[scale=0.9] at (2.2,1.5) [anchor=south west] {$P$};
			\node at (0.1,-0.2) {$g$};
			\draw (P) circle (1.8cm) node (C){};
	\end{tikzpicture}
\end{document}

\documentclass[]{scrartcl} 

\usepackage{tikz} 
\usetikzlibrary{calc} 
\usetikzlibrary{intersections}
\newcommand\Punkt{\tikz[scale=0.07]\draw[thick](-1,-1)--(1,1)(-1,1)--(1,-1);} 


\begin{document} 
\begin{tikzpicture}
	\draw (0,0)--(3.9,0);
		\node at (2.2,1.5) (P) {\Punkt};
		\node[scale=0.9] at (2.2,1.5) [anchor=south west] {$P$};
		\node at (0.1,-0.2) {$g$};
		
	\pgfintersectionofpaths %Dabei, weil ich hoffte, damit die Schnittpunkte extrahieren zu können.
		{\pgfpathcircle{\pgfpoint{2.2cm}{1.5cm}}{1.8cm}}
		{{\pgfpointxy{0}{0}}{\pgfpointxy{3.9cm}{0}}}
		
	\begin{scope}
		\clip  (0,-0.1) rectangle (3.9,0.1);
		\draw (P) circle (1.8cm) node (C){};
	\end{scope}
\end{tikzpicture}
\end{document} 

\documentclass[]{scrartcl}

\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{intersections}
\newcommand\Punkt{\tikz[scale=0.07]\draw[thick](-1,-1)--(1,1)(-1,1)--(1,-1);}

\begin{document}

\begin{tikzpicture}
	\draw[name path=a--b] (0,0)--(3.9,0);
		\node at (2.2,1.5) (P) {\Punkt};
		\node[scale=0.9] at (2.2,1.5) [anchor=south west] {$P$};
		\node at (0.1,-0.2) {$g$};
		
%	\pgfintersectionofpaths
%		{\pgfpathcircle{\pgfpoint{2.2cm}{1.5cm}}{1.8cm}}
%		{{\pgfpointxy{0}{0}}{\pgfpointxy{3.9cm}{0}}}
		
	\begin{scope}
		\clip  (0,-0.1) rectangle (3.9,0.1);
		\draw (P) circle (1.8cm);
	\end{scope}
		\path [name path=c] (P) circle (1.8cm);%Bei dem Befehl darüber erkennt er dann nicht, was "c" ist
	
	\path [name intersections={of=a--b and c}];
	\coordinate [label=below:$A$] (A) at (intersection-1);
	\coordinate [label=below:$B$] (B) at (intersection-2);	

%	\path [name intersections={of=a--b and c, by={[label=below:$A$] (A), [label=below:$B$] (B)}}]; hier hat er Probleme mit den Buchstaben in den runden Klammern, keine Ahnung warum. Die log fragt, ob nicht ein Semikolon fehlt, wüsste aber nicht wo.

\end{tikzpicture}

\end{document}

\documentclass[]{scrartcl} 

\usepackage{tikz} 
\usetikzlibrary{calc} 
\usetikzlibrary{intersections} 
\newcommand\Punkt{\tikz[scale=0.07]\draw[thick](-1,-1)--(1,1)(-1,1)--(1,-1);} 

\begin{document} 

\begin{tikzpicture}
	% Gerade
	\draw[name path=a--b] (0,0)--(3.9,0);
	\node at (0.1,-0.2) {$g$}; 
	% Punkt
	\node at (2.2,1.5) (P) {\Punkt}; 
	\node[scale=0.9] at (2.2,1.5) [anchor=south west] {$P$};
	% Kreisbogen
	\begin{scope} 
		\clip  (0,-0.1) rectangle (3.9,0.1); 
		\draw [name path global=c] (P) circle (1.8cm); 
	\end{scope} 
	% Schnittpunkte
	\path [name intersections={of=a--b and c, by={A, B}}]; 
	\node at (A) [below] {$A$};
	\node at (B) [below] {$B$};
\end{tikzpicture} 

\end{document}