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\documentclass{article}
\usepackage{notebook2e, latexsym}


\begin{document}

Napoleon Punkte (beide)

\dispTFinmath{ \MathBegin{MathArray}{l} \MathBegin{MathArray}{l}
\Mvariable{SetDirectory}[  \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} "D:/Dokumente\multsp und\multsp Einstellungen/vmesser/Eigene\multsp   \\
\noalign{\vspace{0.625ex}}
\hspace{3.em} \Mvariable{Dateien/diplom/Mathematica"}];  \\
\noalign{\vspace{0.625ex}}\\
\MathEnd{MathArray}  \\
\MathBegin{MathArray}{l}
\Mfunction{Needs}["GeoBeweiser`"]  \\
\noalign{\vspace{0.625ex}}  \\
\noalign{\vspace{0.625ex}}\\
\MathEnd{MathArray}  \\
\Muserfunction{clearndg}[];  \\
\noalign{\vspace{0.625ex}}  \\
\MathBegin{MathArray}{l}
\Mfunction{Clear}[\Mvariable{u1},\Mvariable{u2},\Mvariable{u3},\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{x5},\Mvariable{x6},\Mvariable{x7},\Mvariable{x8},
 \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \Mvariable{x9},\Mvariable{x10},\Mvariable{x11},\Mvariable{x12},\Mvariable{x13},\Mvariable{x14},y];\\
\MathEnd{MathArray}
\MathEnd{MathArray}
}

\dispTFinmath{ \MathBegin{MathArray}{l} \MathBegin{MathArray}{l}
\Mvariable{vars}:=\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{x5},\Mvariable{x6},\Mvariable{x7},\Mvariable{x8},\Mvariable{x9},\Mvariable{x10},
 \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Mvariable{x11},\Mvariable{x12},\Mvariable{x13},\Mvariable{x14},y\};  \\
\noalign{\vspace{0.625ex}}\\
\MathEnd{MathArray}  \\
\Mvariable{parameters}:=\{\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\};
\MathEnd{MathArray}
}

\dispTFinmath{ \MathBegin{MathArray}{l}
(*\Mvariable{Punkte}\multsp \Mvariable{definieren}*)  \\
\noalign{\vspace{0.625ex}}\Mfunction{Unprotect}[C];\IndentingNewLine   \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} A=\Muserfunction{point}[0,0];B=\Muserfunction{point}[\Mvariable{u1},0];\IndentingNewLine   \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} C=\Muserfunction{point}[\Mvariable{u2},\Mvariable{u3}];  \\
\noalign{\vspace{0.625ex}}\Mvariable{As}=\Muserfunction{point}[\Mvariable{x1},\Mvariable{x2}];\Mvariable{Bs}=\Muserfunction{point}[\Mvariable{x3},\Mvariable{x4}];
 \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \Mvariable{Cs}=\Muserfunction{point}[\Mvariable{x5},\Mvariable{x6}];\IndentingNewLine   \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} X=\Muserfunction{point}[\Mvariable{x7},\Mvariable{x8}];Y=\Muserfunction{point}[\Mvariable{x9},\Mvariable{x10}];  \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} Z=\Muserfunction{point}[\Mvariable{x11},\Mvariable{x12}];\IndentingNewLine   \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \Mvariable{Napoleon}=\Muserfunction{point}[\Mvariable{x13},\Mvariable{x14}];\\
\MathEnd{MathArray}
}

\dispTFinmath{ \MathBegin{MathArray}{l}
(*\Mvariable{Hilfspunkt}*)  \\
\noalign{\vspace{0.625ex}}\Mvariable{Napoleon}=\Muserfunction{intersectionpoint}[\Muserfunction{ppline}[X,A],  \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Muserfunction{ppline}[Y,B]];\\
\MathEnd{MathArray}
}

\dispTFinmath{ \MathBegin{MathArray}{l}
(*\Mvariable{Konstruiere}\multsp \Mvariable{As}\multsp \Mvariable{Bs}\multsp \Mvariable{Cs}*)  \\
\noalign{\vspace{0.625ex}}\Muserfunction{AddGleichung}[\Mvariable{As}-\Muserfunction{rotate}[C,B,1/3]]\\
\MathEnd{MathArray}
}

\dispTFoutmath{ \Big\{\frac{1}{2}\multsp \big(-\Mvariable{u1}-\Mvariable{u2}-{\sqrt{3}}\multsp \Mvariable{u3}+2\multsp
\Mvariable{x1}\big),\frac{1}{2}\multsp \big(-{\sqrt{3}}\multsp \Mvariable{u1}+{\sqrt{3}}\multsp
\Mvariable{u2}-\Mvariable{u3}+2\multsp \Mvariable{x2}\big)\Big\} }

\dispTFinmath{ \Muserfunction{AddGleichung}[\Mvariable{Bs}-\Muserfunction{rotate}[A,C,1/3]] }

\dispTFoutmath{ \Big\{\frac{1}{2}\multsp \big(-\Mvariable{u2}+{\sqrt{3}}\multsp \Mvariable{u3}+2\multsp
\Mvariable{x3}\big),\frac{1}{2}\multsp \big(-{\sqrt{3}}\multsp \Mvariable{u2}-\Mvariable{u3}+2\multsp
\Mvariable{x4}\big)\Big\} }

\dispTFinmath{ \Muserfunction{AddGleichung}[\Mvariable{Cs}-\Muserfunction{rotate}[B,A,1/3]] }

\dispTFoutmath{ \Big\{\frac{1}{2}\multsp (2\multsp \Mvariable{x5}-\Mvariable{u1}),\frac{1}{2}\multsp
\big({\sqrt{3}}\multsp \Mvariable{u1}+2\multsp \Mvariable{x6}\big)\Big\} }

\dispTFinmath{ \MathBegin{MathArray}{l}
(*\Mvariable{Konstruktion}\multsp \Mvariable{Napoleon}*)  \\
\noalign{\vspace{0.625ex}}\Muserfunction{AddGleichung}[\Muserfunction{centroid}[\Mvariable{As},B,C]-X]\\
\MathEnd{MathArray}
}

\dispTFoutmath{ \Big\{\frac{1}{3}\multsp (\Mvariable{u1}+\Mvariable{u2}+\Mvariable{x1}-3\multsp
\Mvariable{x7}),\frac{1}{3}\multsp (\Mvariable{u3}+\Mvariable{x2}-3\multsp \Mvariable{x8})\Big\} }

\dispTFinmath{ \Muserfunction{AddGleichung}[\Muserfunction{centroid}[A,\Mvariable{Bs},C]-Y] }

\dispTFoutmath{ \Big\{\frac{1}{3}\multsp (\Mvariable{u2}+\Mvariable{x3}-3\multsp \Mvariable{x9}),\frac{1}{3}\multsp
(\Mvariable{u3}-3\multsp \Mvariable{x10}+\Mvariable{x4})\Big\} }

\dispTFinmath{ \Muserfunction{AddGleichung}[\Muserfunction{centroid}[A,B,\Mvariable{Cs}]-Z] }

\dispTFoutmath{ \Big\{\frac{1}{3}\multsp (\Mvariable{u1}-3\multsp \Mvariable{x11}+\Mvariable{x5}),\frac{1}{3}\multsp
(\Mvariable{x6}-3\multsp \Mvariable{x12})\Big\} }

\dispTFinmath{ \Mvariable{polys}=\Mfunction{Map}[\Mfunction{Numerator}[\Mfunction{Together}[\#]]\multsp \&,\multsp
\Mvariable{Gleichungen}] }

\dispTFoutmath{ \MathBegin{MathArray}{l} \big\{-\Mvariable{u1}-\Mvariable{u2}-{\sqrt{3}}\multsp \Mvariable{u3}+2\multsp
\Mvariable{x1},-{\sqrt{3}}\multsp \Mvariable{u1}+{\sqrt{3}}\multsp
\Mvariable{u2}-\Mvariable{u3}+2\multsp \Mvariable{x2},  \\
\noalign{\vspace{0.6875ex}}
\hspace{1.em} -\Mvariable{u2}+{\sqrt{3}}\multsp \Mvariable{u3}+2\multsp \Mvariable{x3},-{\sqrt{3}}\multsp \Mvariable{u2}-\Mvariable{u3}+2\multsp
\Mvariable{x4},2\multsp \Mvariable{x5}-\Mvariable{u1},  \\
\noalign{\vspace{0.6875ex}}
\hspace{1.em} {\sqrt{3}}\multsp \Mvariable{u1}+2\multsp \Mvariable{x6},\Mvariable{u1}+\Mvariable{u2}+\Mvariable{x1}-3\multsp \Mvariable{x7},\Mvariable{u3}+\Mvariable{x2}-3\multsp
\Mvariable{x8},  \\
\noalign{\vspace{0.75ex}}
\hspace{1.em} \Mvariable{u2}+\Mvariable{x3}-3\multsp \Mvariable{x9},\Mvariable{u3}-3\multsp \Mvariable{x10}+\Mvariable{x4},\Mvariable{u1}-3\multsp
\Mvariable{x11}+\Mvariable{x5},\Mvariable{x6}-3\multsp \Mvariable{x12}\big\}\\
\MathEnd{MathArray}
}

\dispTFinmath{
\Mvariable{con}=\Mfunction{Numerator}[\Muserfunction{online}[\Mvariable{Napoleon},\Muserfunction{ppline}[Z,C]]] }

\dispTFoutmath{ \MathBegin{MathArray}{l} \Mvariable{u1}\multsp \Mvariable{u3}\multsp \Mvariable{x10}\multsp
\Mvariable{x7}-\Mvariable{u3}\multsp \Mvariable{x10}\multsp \Mvariable{x11}\multsp \Mvariable{x7}-\Mvariable{u1}\multsp
\Mvariable{x10}\multsp \Mvariable{x12}\multsp \Mvariable{x7}+\Mvariable{u2}\multsp \Mvariable{x10}\multsp
\Mvariable{x12}\multsp
\Mvariable{x7}-\Mvariable{u1}\multsp \Mvariable{u2}\multsp \Mvariable{x10}\multsp \Mvariable{x8}-  \\
\noalign{\vspace{0.625ex}}
\hspace{1.em} \Mvariable{u1}\multsp \Mvariable{u3}\multsp \Mvariable{x11}\multsp \Mvariable{x8}+\Mvariable{u1}\multsp \Mvariable{x10}\multsp \Mvariable{x11}\multsp
\Mvariable{x8}+\Mvariable{u1}\multsp \Mvariable{u2}\multsp \Mvariable{x12}\multsp \Mvariable{x8}+\Mvariable{u3}\multsp \Mvariable{x11}\multsp \Mvariable{x8}\multsp
\Mvariable{x9}-\Mvariable{u2}\multsp \Mvariable{x12}\multsp \Mvariable{x8}\multsp \Mvariable{x9}\\
\MathEnd{MathArray}
}

\dispTFinmath{ \Muserfunction{IdealMember}[\Mvariable{con},\Mvariable{polys},\Mvariable{vars}]//\Mfunction{Timing} }

\dispTFoutmath{ \{0.2\multsp \Mvariable{Second},\Mvariable{True}\} }

\dispTFinmath{ \Mvariable{con1}=\Mfunction{Numerator}[\Muserfunction{sqrdist}[X,Y]-\Muserfunction{sqrdist}[Y,Z]] }

\dispTFoutmath{ -{{\Mvariable{x11}}^2}+2\multsp \Mvariable{x9}\multsp
\Mvariable{x11}-{{\Mvariable{x12}}^2}+{{\Mvariable{x7}}^2}+{{\Mvariable{x8}}^2}+2\multsp \Mvariable{x10}\multsp
\Mvariable{x12}-2\multsp \Mvariable{x10}\multsp \Mvariable{x8}-2\multsp \Mvariable{x7}\multsp \Mvariable{x9} }

\dispTFinmath{ \Mvariable{con2}=\Mfunction{Numerator}[\Muserfunction{sqrdist}[X,Z]-\Muserfunction{sqrdist}[Y,Z]] }

\dispTFoutmath{ -{{\Mvariable{x10}}^2}+2\multsp \Mvariable{x12}\multsp
\Mvariable{x10}+{{\Mvariable{x7}}^2}+{{\Mvariable{x8}}^2}-{{\Mvariable{x9}}^2}-2\multsp \Mvariable{x11}\multsp
\Mvariable{x7}-2\multsp \Mvariable{x12}\multsp \Mvariable{x8}+2\multsp \Mvariable{x11}\multsp \Mvariable{x9} }

\dispTFinmath{ \Muserfunction{IdealMember}[\Mvariable{con1},\Mvariable{polys},\Mvariable{vars}]//\Mfunction{Timing} }

\dispTFoutmath{ \{0.06\multsp \Mvariable{Second},\Mvariable{True}\} }

\dispTFinmath{ \Muserfunction{IdealMember}[\Mvariable{con2},\Mvariable{polys},\Mvariable{vars}]//\Mfunction{Timing} }

\dispTFoutmath{ \{0.04\multsp \Mvariable{Second},\Mvariable{True}\} }

Beweis der Bemerkung: Der Napoleonpunkt liegt auf der Eulergeraden.

\dispTFinmath{ \Mvariable{Schwerpunkt}=\Muserfunction{centroid}[A,B,C] }

\dispTFoutmath{ \Big\{\frac{\Mvariable{u1}+\Mvariable{u2}}{3},\frac{\Mvariable{u3}}{3}\Big\} }

\dispTFinmath{ \Mvariable{Umkreismittelpunkt}=\Muserfunction{circumcenter}[A,B,C] }

\dispTFoutmath{ \Big\{\frac{\Mvariable{u1}}{2},\frac{{{\Mvariable{u2}}^2}-\Mvariable{u1}\multsp
\Mvariable{u2}+{{\Mvariable{u3}}^2}}{2\multsp \Mvariable{u3}}\Big\} }

\dispTFinmath{ \MathBegin{MathArray}{l}
\Mvariable{con3}=\Muserfunction{online}[\Mvariable{Napoleon},  \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Muserfunction{ppline}[\Mvariable{Schwerpunkt},\Mvariable{Umkreismittelpunkt}]]\\
\MathEnd{MathArray}
}

\dispTFoutmath{ \MathBegin{MathArray}{l} (\Mvariable{u2}\multsp \Mvariable{x8}\multsp {{\Mvariable{u1}}^3}-2\multsp
\Mvariable{u2}\multsp \Mvariable{x10}\multsp \Mvariable{x7}\multsp {{\Mvariable{u1}}^2}-\Mvariable{u3}\multsp
\Mvariable{x10}\multsp \Mvariable{x8}\multsp {{\Mvariable{u1}}^2}-  \\
\noalign{\vspace{0.601563ex}}
\hspace{3.em} \Mvariable{u2}\multsp \Mvariable{x8}\multsp \Mvariable{x9}\multsp {{\Mvariable{u1}}^2}+3\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{x10}\multsp
\Mvariable{x7}\multsp \Mvariable{u1}+{{\Mvariable{u3}}^2}\multsp \Mvariable{x10}\multsp \Mvariable{x7}\multsp \Mvariable{u1}-{{\Mvariable{u2}}^3}\multsp
\Mvariable{x8}\multsp \Mvariable{u1}-  \\
\noalign{\vspace{0.601563ex}}
\hspace{3.em} \Mvariable{u2}\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x8}\multsp \Mvariable{u1}+2\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp
\Mvariable{x10}\multsp \Mvariable{x8}\multsp \Mvariable{u1}-{{\Mvariable{u2}}^3}\multsp \Mvariable{x10}\multsp \Mvariable{x7}-\Mvariable{u2}\multsp
{{\Mvariable{u3}}^2}\multsp \Mvariable{x10}\multsp \Mvariable{x7}+  \\
\noalign{\vspace{0.601563ex}}
\hspace{3.em} {{\Mvariable{u2}}^3}\multsp \Mvariable{x8}\multsp \Mvariable{x9}+\Mvariable{u2}\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x8}\multsp
\Mvariable{x9})/(\Mvariable{x10}\multsp \Mvariable{x7}+\Mvariable{u1}\multsp \Mvariable{x8}-\Mvariable{x8}\multsp \Mvariable{x9})\\
\MathEnd{MathArray}
}

\dispTFinmath{ \Muserfunction{IdealMember}[\Mvariable{con3},\Mvariable{polys},\Mvariable{vars}]//\Mfunction{Timing} }

\dispTFoutmath{ \{0.231\multsp \Mvariable{Second},\Mvariable{True}\} }


\end{document}
