MP: * Gleichseitige Dreiecke und Gerade (Forum Matroids Matheplanet)

2021-01-13 04:47 - Wario im Themenstart schreibt:
Es seien $n\geq 2$ gleichseitige Dreiecke der Kantenlänge $s$ wie in folgender Abbildung nebeneinander angeordnet, die, wie gezeigt, von einer Geraden geschnitten werden:

$<math> \def\allgm{1} \def\bew{0} \pagecolor{black!1} % Gegebene Gr��en \pgfmathsetmacro{\N}{5} \pgfmathsetmacro{\s}{2} \colorlet{unten}{orange!55} \colorlet{oben}{yellow!55} %\ifnum\bew=1 \pgfmathsetmacro{\N}{\bew==1 ? 7 : 6} %\fi \pgfmathsetmacro{\Ns}{int(\N+1)} \pgfmathsetmacro{\Np}{int(\N-1)} % Annotation - Kanten \ifnum\bew=1 \tikzset{kante/.style={text=black} } \else \tikzset{kante/.style={text=black!1} } \fi % Layers \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground} \begin{tikzpicture}[%scale=0.7, font=\footnotesize, background rectangle/.style={draw=none, fill=black!1, rounded corners}, %show background rectangle, Dreieck/.style={}, every label/.style={text=black!1}, ] \foreach[count=\nI] \n in {1,...,\N}{ \begin{scope}[xshift=-\nI*\s cm] \coordinate[label=below:$A\n$] (A\n) at (0,0); \coordinate[label=below:$B\n$] (B\n) at (\s,0); \coordinate[label=$C\n$] (C\n) at (60:\s); % unten f�llen \draw[Dreieck, name path global=dreieck\n] (A\n) -- (B\n) -- (C\n) --cycle; % Annotation unten \ifnum\bew=1 \path[] (A\n) -- (B\n) node[midway, below]{$s$}; \fi \end{scope} } % Annotation rechts \ifnum\bew=1 \path[] (B1) -- (C1) node[midway, right]{$s$}; \fi % Gerade 1/2 \path[name path global=gerade] (C1) -- (A\N); \foreach[count=\kNo] \k in {2,...,\N}{ \pgfmathtruncatemacro\K{\k+1} \path[name intersections={of={dreieck\k} and gerade, name=Point}]; \ifnum\k=\N \begin{pgfonlayer}{background} \coordinate[label=X\k] (X\k) at (Point-2); \coordinate[label=Y\k] (Y\k) at (Point-1); \path[] (X\k) -- (C\k) node[kante, midway, right]{$x_{\k}$}; \path[] (Y\k) -- (C\k) node[kante, midway, left]{$s$ %$y_{\k}$ }; \end{pgfonlayer} \else \begin{pgfonlayer}{background} \coordinate[label=X\k] (X\k) at (Point-1); \coordinate[label=Y\k] (Y\k) at (Point-2); \path[] (X\k) -- (C\k) node[kante, midway, right]{$x_{\k}$}; \path[] (Y\k) -- (C\k) node[kante, midway, left]{$y_{\k}$}; \end{pgfonlayer} \fi % oben f�llen \ifnum\bew=1 \begin{pgfonlayer}{background} \fill[oben] (X\k) -- (Y\k) -- (C\k) --cycle; \end{pgfonlayer} \else \begin{pgfonlayer}{background} \fill[unten] (X\k) -- (B\k) -- (A\k) -- (Y\k) --cycle; \fill[unten] (A1) -- (B1) -- (C1) --cycle; \end{pgfonlayer} \fi } % Gerade 2/2 \draw[Dreieck] (C1) -- (A\N); % Beweis ======================== % Annotation Strahlensatzfigur \ifnum\bew=1%========================== \draw[densely dashed] (C1) -- (C\N) -- +(-\s,0) coordinate[label=$C\Ns$] (C\Ns) node[midway, above]{$$} -- (A\N) node[midway, left]{$s$}; % % Annotation oben \foreach[evaluate={\K=int(\k+1)}] \k in {1,...,\N}{ \pgfmathsetmacro\text{\k==3 || \k==5 ? "" : "s"} \path[] (C\k) -- (C\K) node[midway, above]{$\text$}; } \foreach \k/\text in {\Ns/{n+1},\N/{n},\Np/{n-1}}{ \node[above] at (C\k) {$P_{\text}$}; } \foreach \k in {1,2,3}{ \node[above] at (C\k) {$P_{\k}$}; } \fi%============================ % Allgemeine Zeichnung \ifnum\allgm=1 \pgfmathtruncatemacro{\Na}{\N-2} \pgfmathtruncatemacro{\Nb}{3} \fill[black!1] ([shift={(0,-2em)}]A\Na) rectangle ([shift={(-0.5*\s cm,2em)}]C\Nb) node[midway, text=black, font=\Huge]{$\boldsymbol\dots$}; \fi \end{tikzpicture} </math>$

Berechne die hervorgehobene Gesamtfläche.

Lösung.

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$<math> \def\allgm{0} \def\bew{0} \pagecolor{black!1} % Gegebene Gr��en \pgfmathsetmacro{\N}{7} \pgfmathsetmacro{\s}{2} \colorlet{unten}{orange!66} \colorlet{oben}{yellow!66} \pgfmathsetmacro{\N}{\bew==1 ? 7 : \N} \pgfmathsetmacro{\Ns}{int(\N+1)} \pgfmathsetmacro{\Np}{int(\N-1)} % Annotation - Kanten \ifnum\bew=1 \tikzset{kante/.style={text=black} } \else \tikzset{kante/.style={text=black!1} } \fi % Layers \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground} \begin{tikzpicture}[%scale=0.7, font=\footnotesize, background rectangle/.style={draw=none, fill=black!1, rounded corners}, %show background rectangle, Dreieck/.style={}, every label/.style={text=black!1}, ] \foreach[count=\nI] \n in {1,...,\N}{ \begin{scope}[xshift=-\nI*\s cm] \coordinate[label=below:$A\n$] (A\n) at (0,0); \coordinate[label=below:$B\n$] (B\n) at (\s,0); \coordinate[label=$C\n$] (C\n) at (60:\s); % unten f�llen \draw[Dreieck, name path global=dreieck\n] (A\n) -- (B\n) -- (C\n) --cycle; % Annotation unten \ifnum\bew=1 \path[] (A\n) -- (B\n) node[midway, below]{$s$}; \fi \end{scope} } % Annotation rechts \ifnum\bew=1 \path[] (B1) -- (C1) node[midway, right]{$s$}; \fi % Gerade 1/2 \path[name path global=gerade] (C1) -- (A\N); \foreach[count=\kNo] \k in {2,...,\N}{ \pgfmathtruncatemacro\K{\k+1} \path[name intersections={of={dreieck\k} and gerade, name=Point}]; \ifnum\k=\N \begin{pgfonlayer}{background} \coordinate[label=X\k] (X\k) at (Point-2); \coordinate[label=Y\k] (Y\k) at (Point-1); \path[] (X\k) -- (C\k) node[kante, midway, right]{$x_{n}$}; \path[] (Y\k) -- (C\k) node[kante, midway, left]{$s$ %$y_{\k}$ }; \end{pgfonlayer} \else \begin{pgfonlayer}{background} \coordinate[label=X\k] (X\k) at (Point-1); \coordinate[label=Y\k] (Y\k) at (Point-2); \ifnum\k=\Np \path[] (X\k) -- (C\k) node[kante, pos=0.7, right]{$x_{n-1}$}; \path[] (Y\k) -- (C\k) node[kante, pos=0.7, left]{$y_{n-1}$}; \else \path[] (X\k) -- (C\k) node[kante, midway, right]{$x_{\k}$}; \path[] (Y\k) -- (C\k) node[kante, midway, left]{$y_{\k}$}; \fi \end{pgfonlayer} \fi % oben f�llen \ifnum\bew=1 \begin{pgfonlayer}{background} \fill[oben] (X\k) -- (Y\k) -- (C\k) --cycle; \end{pgfonlayer} \else \begin{pgfonlayer}{background} \fill[unten] (X\k) -- (B\k) -- (A\k) -- (Y\k) --cycle; \fill[unten] (A1) -- (B1) -- (C1) --cycle; \end{pgfonlayer} \fi } % Gerade 2/2 \draw[Dreieck] (C1) -- (A\N); % Beweis ======================== % Annotation Strahlensatzfigur \ifnum\bew=1%========================== \draw[densely dashed] (C1) -- (C\N) -- +(-\s,0) coordinate[label=$C\Ns$] (C\Ns) node[midway, above]{$$} -- (A\N) node[midway, left]{$s$}; % % Annotation oben \foreach[evaluate={\K=int(\k+1)}] \k in {1,...,\N}{ \pgfmathsetmacro\text{\k==3 || \k==5 ? "" : "s"} \path[] (C\k) -- (C\K) node[midway, above]{$\text$}; } \foreach \k/\text in {\Ns/{n+1},\N/{n},\Np/{n-1}}{ \node[above] at (C\k) {$P_{\text}$}; } \foreach \k in {1,2,3}{ \node[above] at (C\k) {$P_{\k}$}; } \fi%============================ % Allgemeine Zeichnung \ifnum\allgm=1%===================== \pgfmathtruncatemacro{\Na}{\N-2} \pgfmathtruncatemacro{\Nb}{3} \fill[black!1] ([shift={(0,-2em)}]A\Na) rectangle ([shift={(-0.5*\s cm,2em)}]C\Nb) node[midway, text=black, font=\Huge]{$\boldsymbol\dots$}; \fi%===================== \end{tikzpicture} </math>$

$<math> \pgfmathsetlengthmacro{\u}{0.567cm} \begin{tikzpicture}[%scale=0.7, font=\footnotesize, background rectangle/.style={draw=none, fill=black!1, rounded corners}, show background rectangle, y={(\u,0cm)}, x={({0.3*\u}, {-0.45*\u})}, z={(0cm,\u)}, every label/.style={text=black!1}, ] % Ansicht \def\vgl{1} % Anzahl Stufen \pgfmathsetmacro{\N}{5} % Einstellungen \def\vglvol{0} \def\abschlusspyramide{1} \def\vglwuerfel{1} \def\cosy{0} \colorlet{wuerfel}{lightgray} \colorlet{Prisma}{green!77!black} \colorlet{prisma}{Prisma!44} \colorlet{pyramide}{red!55} \ifnum\vgl=0 \def\abschlusspyramide{0} \def\vglvol{0} \def\vglwuerfel{1} \else \def\abschlusspyramide{1} \def\vglwuerfel{0} \fi %\pgfmathsetmacro\k{4} % Layers \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground} % Blockpyramide \foreach[count=\K from 0] \k in {\N,...,1}{%========================== \pgfmathsetmacro{\kp}{\k-1} \begin{scope}[shift={(0,0,\K)}, local bounding box=blockpyramide] \begin{pgfonlayer}{background} \coordinate[label=left:$A\k$] (A\k) at (0,0,0); \coordinate[label=left:$B\k$] (B\k) at (\k,0,0); \coordinate[label=right:$C\k$] (C\k) at (\k,\k,0); \coordinate[label=right:$D\k$] (D\k) at (0,\k,0); \coordinate[label=left:$A\k"$] (A\k1) at (0,0,1); \coordinate[label=left:$B\k"$] (B\k1) at (\k,0,1); \coordinate[label=right:$C\k"$] (C\k1) at (\k,\k,1); \coordinate[label=right:$D\k"$] (D\k1) at (0,\k,1); \end{pgfonlayer} % Grund- und Deckfl�che \draw[] (A\k1) -- (B\k1) -- (C\k1) -- (D\k1) --cycle; \draw[] (A\k) -- (B\k) -- (C\k) ; \foreach \P in {A\k,B\k,C\k} \draw[] (\P) -- (\P1); % F�llung der W�rfelfl�chen 1/3 \foreach \x in {0,...,\kp}{ \foreach \y in {0,...,\kp}{ \fill[wuerfel] ([shift={(\x,\y,0)}]A\k1) -- ++(1,0,0) -- ++(0,1,0) -- ++(-1,0,0) --cycle; }} \foreach \x in {0,...,\kp}{ % F�llung der W�rfelfl�chen 2/3 \fill[wuerfel] ([shift={(\x,0,0)}]A\k) -- ++(1,0,0) -- ++(0,0,1) -- ++(-1,0,0) --cycle; } \foreach \y in {0,...,\kp}{ % F�llung der W�rfelfl�chen 3/3 \fill[wuerfel] ([shift={(0,\y,0)}]B\k) -- ++(0,1,0) -- ++(0,0,1) -- ++(0,-1,0) --cycle; % Hilfslinien 2/2 \draw[] ([shift={(0,\y,0)}]B\k) -- ++(0,0,1) -- ++(-\k,0,0); } % Hilfslinien 2/2 \foreach \x in {0,...,\k}{ \draw[] ([shift={(\x,0,0)}]A\k) -- ++(0,0,1) -- ++(0,\k,0); } \ifnum\vgl=1%============================== % Kantenpyramiden rechts \draw[Prisma] (C\k) -- ++(0,1,0) coordinate(P) -- ++(-\k,0,0) coordinate(Q) -- (D\k1) coordinate(R) -- (C\k1) coordinate(S) --(P); \fill[prisma] (P) -- (Q) -- (R) -- (S) --cycle; % Eckpyramiden \draw[red] (C\k) -- ++(0,1,0) coordinate(P) -- ++(1,0,0) coordinate(Q) -- ++(0,-1,0) coordinate(R) --cycle; \path[draw=red, fill=pyramide] (P) -- (Q) -- (C\k1) --cycle; \path[draw=red, fill=pyramide] (Q) -- (R) -- (C\k1) --cycle; % Kantens�ulen vorne \draw[Prisma] (C\k) -- ++(1,0,0) coordinate(P) -- ++(0,-\k,0) coordinate(Q) -- ++(-1,0,0) coordinate(R) -- ++(0,0,1) coordinate(S) -- (C\k1) coordinate(T); \fill[prisma] (P) -- (Q) -- (S) -- (T) --cycle; \fill[prisma] (Q) -- (R) -- (B\k1) --cycle; \draw[Prisma] (P) -- (Q); \draw[Prisma] (Q) -- (S); \draw[Prisma] (P) -- (T); \fi%============================== \end{scope} }%========================== % Abschlu�pyramide oben \ifnum\abschlusspyramide=1 \path[] (A11) -- +(0,0,1) coordinate(S); \path[draw=red, fill=pyramide] (S) -- (B11) -- (C11) --cycle; \path[draw=red, fill=pyramide] (S) -- (A11) -- (B11) --cycle; \path[draw=red, fill=pyramide] (S) -- (C11) -- (D11) --cycle; \fi \ifnum\cosy=1 \begin{scope}[-latex, shift={(0,3,4)}] \foreach \P/\s/\Pos in {(1,0,0)/x/below, (0,1,0)/y/above, (0,0,1)/z/right} \draw[] (0,0,0) -- \P node[\Pos, pos=0.9,inner sep=2pt]{$\s$}; \end{scope} \fi % Vergleichsk�rper \ifnum\vglvol=1%=========================== \begin{scope}[shift={($(blockpyramide.north east)+(0,\N-1,-2)$)}] \draw[Prisma] (0,0,0) coordinate(P) -- (2,0,0) coordinate(Q) node[near start, below]{1} -- ++(0,3,0) coordinate(R) node[midway, below]{$k$} -- ++(-2,0,0) coordinate(S) -- cycle; \draw[Prisma] (0,0,2) coordinate(Ps) -- (0,3,2) coordinate(Ss); \fill[prisma] (Q) -- (R) -- (Ss) -- (Ps) --cycle; \fill[prisma] (P) -- (Q) -- (Ps) --cycle; \draw[Prisma] (Ps) -- (P) node[midway, left]{1} (Ps) -- (Q); \draw[Prisma] (Ss) -- (R); \node[anchor=west] at ($(R)!0.5!(Ss)$) {$ V_\text{KS}(k) =G h = \frac12 \cdot 1 \cdot 1 \cdot k = \underline{ \frac12 k =V_\text{KS}(k) }$}; \end{scope} \begin{scope}[shift={($(blockpyramide.north east)+(0,\N-0,-5.75)$)}] \coordinate(S) at (0,0,2); \coordinate(A) at (0,0,0); \coordinate(B) at (2,0,0); \coordinate(C) at (2,2,0); \coordinate(D) at (0,2,0); \draw[red] (A) -- (B) node[near start, below]{1} -- (C) node[midway, below]{1} -- (D) --cycle; \path[draw=red, fill=pyramide] (S) -- (B) -- (C) --cycle; \path[draw=red, fill=pyramide] (S) -- (A) node[red, midway, left]{1} -- (B) --cycle; \path[draw=red, fill=pyramide] (S) -- (C) -- (D) --cycle; \node[anchor=west, yshift=-2cm] at ($(R)!0.5!(Ss)$) {$ V_\text{EPy0}=\frac13 G h = \frac13 \cdot 1 \cdot 1 = \underline{ \frac13 =V_\text{EPy0} }$}; \end{scope} \fi%=========================== \ifnum\vglwuerfel=1%=========================== \pgfmathsetmacro\e{1.3} \begin{scope}[shift={($(blockpyramide.north east)+(0,\N-1,-2)$)}] \coordinate(A) at (0,0,0); \coordinate(B) at (\e,0,0); \coordinate(C) at (\e,\e,0); \coordinate(D) at (0,\e,0); \coordinate(A1) at (0,0,\e); \coordinate(B1) at (\e,0,\e); \coordinate(C1) at (\e,\e,\e); \coordinate(D1) at (0,\e,\e); \draw[] (A) -- (B) node[near start, below]{1} -- (C) node[midway, below]{1} -- (D) --cycle; \fill[wuerfel] (A) -- (B) -- (B1) -- (A1) --cycle; \fill[wuerfel] (B) -- (C) -- (C1) -- (B1) --cycle; \fill[wuerfel] (A1) -- (B1) -- (C1) -- (D1) --cycle; \draw[] (A1) -- (B1) -- (C1) -- (D1) --cycle; \path[] (A1) -- (A) node[midway, left]{1}; \foreach \P in {A,B,C} \draw[] (\P) -- (\P1); \node[anchor=west] at ($(C)!0.5!(C1)$) {$ V_\text{W} =a^3 = 1^3 = \underline{ 1 =V_\text{W} }$}; \end{scope} \fi%=========================== \end{tikzpicture} </math>$