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\begin{document}
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\noindent\textbf{Mathematisches Institut} \hfill \textbf{Elementargeometrie}\\
Andriy Haydys \hfill Sommersemester 2020\\
Giorgio Laguzzi\\

\vspace{5mm}

\mytitle{\begin{center}Übungsblatt 1\end{center}}
\begin{center}Abgabe bis 18.5.2020, 12:00 per E-Mail an Ihren Tutor\end{center}


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Sie d\"urfen f\"ur Übung 1 und 2 alle Propositionen aus Buch I des Euklids \emph{Elements} verwenden.
Für Übung 3 und 4 dürfen Sie zusätzlich die Propositionen aus Buch III verwenden.





\textbf{\"Ubung 1. (4 Punkte)} 

Seien $\Delta abc$ ein Dreieck und $\beta_1, \beta_2, \beta_3$ die drei Mittelsenkrechten.  Zeigen Sie, dass sich $\beta_1, \beta_2, \beta_3$ in einem eindeutigen Punkt schneiden.
\begin{center}
\begin{tikzpicture}
\draw (0,0) node[anchor=north east]{$a$} -- (3,0) node[anchor=north west]{$b$} ;
\draw (0,0) -- (2,2) node[anchor=south]{$c$} -- (3,0);
\draw (1,1) -- (2.9,-1) node[anchor=north east]{$\beta_1$};
\draw (2.5,1) -- (-1.3,-1) node[anchor=north west]{$\beta_2$};
\draw (1.5,0) -- (1.5,2.5) node[anchor=east]{$\beta_3$};
\end{tikzpicture}
\end{center}






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\textbf{\"Ubung 2. (4 Punkte)}
Betrachten Sie ein Quadrat $abcd$ und einen Punkt $p$ auf der Seite $cd$. Zeigen Sie, dass die Summe die Seiten des Dreiecks $\Delta abp$ streng größer als die Summe der zwei Diagonalen des Quadrats $abcd$ ist (d.h, $ab + bp + pa > ac + bd$).  
\begin{center}
\begin{tikzpicture}
\draw (0,0) node[anchor=north east]{$a$} -- (2,0) node[anchor=north west]{$b$}-- (2,2) node[anchor=south west]{$c$};
\draw  (2,2) -- (0,2)node[anchor=south east]{$d$} -- (0,0);
\draw (0,0) -- (1.5,2) node[anchor=south]{$p$} -- (2,0);
\draw[-,dotted]   (0,0) --  (2,2);
\draw[-,dotted]   (2,0) --  (0,2);
\end{tikzpicture}
\end{center}

\vspace{5mm}
 
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\textbf{\"Ubung 3. ($\mathbf{1+1+2}$ Punkte)} 
Skizzieren Sie die folgenden Konstruktionen mit Zirkel und Lineal. Es sind keine formellen Nachweise erforderlich, aber bitte geben Sie eine kurze intuitive
Erklärung an, warum jede Konstruktion funktioniert.
\begin{itemize}
\item (1 Punkt) Gegeben ein Kreis, finden Sie den Mittelpunkt.
\item (1 Punkte) Gegeben ein Kreis $C$ und ein externer Punkt $p$, zeichnen Sie die Tangente $g$ von $C$, die den Punkt $p$ enth\"alt.
\item (2 Punkte) Gegeben eine Gerade $g$, ein Punkt $p \in g$ und ein anderer Punkt $q \notin g$, Konstruieren Sie einen Kreis $C$ sodass $p,q \in C$ und $g$ Tangente von $C$ ist.
\end{itemize}

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\textbf{\"Ubung 4. ($\mathbf{2+2}$ Puntke)}
\begin{itemize}
\item (2 Punkte) Gegeben ein Rechteck, Konstruieren Sie ein Quadrat mit gleichem Inhalt. (\emph{Hilfe}: Verwenden Sie III.35 in Euklids.) 
\item (2 Punkte) Gegeben ein Gerade $g$ und zwei Punkte $p,q$, konstruiren Sie ein Kreis $C$ den die Punkte $p,q$ enth\"alt, so dass $g$ ein Tangente von $C$ ist.
(\emph{Hilfe}: Verwenden Sie III.37 in Euklids.)
\end{itemize}




\end{document}





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