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Kein bestimmter Bereich J Bezeichnung der Winkel am Dreieck aus den Seiten-Schnittpunkten mit den Winkelhalbierenden
Wario
Aktiv Letzter Besuch: in der letzten Woche
Dabei seit: 01.05.2020
Mitteilungen: 947
  Themenstart: 2022-07-03

Wie könnte ich die vier Winkel an $W_a, W_b,W_c$ sinnvoll und systematisch bezeichnen? Am Punkt $W_b$ bzw. $W_a$ sind zur Zeit nur zwei von vier Winkeln eingezeichnet - mit vorläufigen Bezeichnungen. Bei $W_c$ ist noch nichts eingezeichnet. Erster eigener Versuch: $% Seitenlängen \pgfmathsetmacro{\a}{5} % 4 \pgfmathsetmacro{\b}{4.5} % 3.5 \pgfmathsetmacro{\c}{7} % 6 \pgfmathsetmacro{\Alpha}{acos((\b^2+\c^2-\a^2)/(2*\b*\c))} \pgfmathsetmacro{\Beta}{acos((\a^2+\c^2-\b^2)/(2*\a*\c))} \pgfmathsetmacro{\Gamma}{acos((\a^2+\b^2-\c^2)/(2*\a*\b))} \pgfmathsetmacro{\wA}{2*\b*\c*cos(\Alpha/2)/(\b+\c)} \pgfmathsetmacro{\wB}{2*\a*\c*cos(\Beta/2)/(\a+\c)} \pgfmathsetmacro{\wC}{2*\a*\b*cos(\Gamma/2)/(\a+\b)} % Inkreis \pgfmathsetmacro{\s}{0.5*(\a+\b+\c)} \pgfmathsetmacro{\F}{sqrt(\s*(\s-\a)*(\s-\b)*(\s-\c))} \pgfmathsetmacro{\r}{\F/\s} \pgfmathsetmacro{\ai}{\a/(2*\s)} \pgfmathsetmacro{\bi}{\b/(2*\s)} \pgfmathsetmacro{\ci}{\c/(2*\s)} \pgfkeys{/tikz/savevalue/.code 2 args={\global\edef#1{#2}}} \begin{tikzpicture}[%scale=0.7, font=\footnotesize, background rectangle/.style={draw=none, fill=black!1, rounded corners}, show background rectangle, Punkt/.style 2 args={ label={[#1]:$#2$} }, Dreieck/.style={thick}, ] % Dreieckskonstruktion \coordinate[label=below:$A$] (A) at (0,0); \coordinate[label=below:$B$] (B) at (\c,0); \coordinate[label=$C$] (C) at (\Alpha:\b); \draw[local bounding box=dreieck] (A) -- (B) -- (C) --cycle; % Winkelhalbierende \draw[] (A) --+ (0.5*\Alpha:\wA) coordinate[label={[inner sep=1pt]45:$W_a$}] (Wa) node[pos=0.285, sloped, above, inner sep=1pt]{$w_\alpha$}; \path[] (B) -- (Wa) node[midway, right]{$a_B$}; \path[] (C) -- (Wa) node[near start, right]{$a_C$}; \draw[] (B) --+ (180-0.5*\Beta:\wB) coordinate[label={[inner sep=1pt]135:$W_b$}] (Wb) node[pos=0.3125, sloped, above, inner sep=1pt]{$w_\beta$}; \path[] (A) -- (Wb) node[midway, left]{$b_A$}; \path[] (C) -- (Wb) node[near start, left]{$b_C$}; \draw[] (C) --+ (-\Beta-0.5*\Gamma:\wC) coordinate[label=below:$W_c$] (Wc) node[pos=0.25, right, inner sep=1pt]{$w_\gamma$}; \path[] (A) -- (Wc) node[midway, below]{$c_A$}; \path[] (B) -- (Wc) node[midway, below]{$c_B$}; % Halbe Winkel \draw pic [draw, angle radius=12mm, angle eccentricity=0.8, % pic text={$\alpha$}, pic text options={}, "$\frac{\alpha}{2}$" ] {angle =B--A--Wa}; \draw pic [draw, angle radius=12mm, angle eccentricity=0.8, % pic text={$\alpha$}, pic text options={}, "$\frac{\alpha}{2}$" ] {angle =Wa--A--C}; \draw pic [draw, angle radius=15mm, angle eccentricity=0.9125, % pic text={$\alpha$}, pic text options={}, "$\frac{\beta}{2}$" ] {angle =C--B--Wb}; \draw pic [draw, angle radius=15mm, angle eccentricity=0.93, % pic text={$\alpha$}, pic text options={}, "$\frac{\beta}{2}$" ] {angle =Wb--B--A}; \draw pic [draw, angle radius=6mm, angle eccentricity=0.7, % pic text={$\alpha$}, pic text options={}, "$\frac{\gamma}{2}$" ] {angle =A--C--Wc}; \draw pic [draw, angle radius=6mm, angle eccentricity=0.7, % pic text={$\alpha$}, pic text options={}, "$\frac{\gamma}{2}$" ] {angle =Wc--C--B}; % Inkreis \coordinate[label={[inner sep=1.75pt]75:$I$}] (I) at ($\ai*(A)+\bi*(B)+\ci*(C)$); %\draw[] (I) circle[radius=\r]; % Winkelhalbierenden-Dreieck \draw[local bounding box=dreieck] (Wa) -- (Wb) -- (Wc) --cycle; % Winkel am Winkelhalbierenden-Dreieck \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\delta_a$" ] {angle =C--Wa--Wb}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\varepsilon_b$" ] {angle =Wb--Wa--A}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\delta_b$" ] {angle =Wa--Wb--C}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\varepsilon_b$" ] {angle =B--Wb--Wa}; %\path[] (A) -- (C) node[midway, left]{$b$}; %\path[] (B) -- (C) node[midway, right]{$a$}; %\draw pic [draw, angle radius=4mm, %angle eccentricity=1.8, % pic text={$\alpha$}, pic text options={}, %"$\chi$", %] {angle =C--W--A}; %\draw pic [draw, angle radius=5mm, %angle eccentricity=1.8, % pic text={$\alpha$}, pic text options={}, %"$\overline{\chi}$", %] {angle =B--W--C}; %% Punkte \foreach \P in {Wa, Wb, Wc, I} \draw[fill=black!1] (\P) circle (1.5pt); %% Annottion - Text %\node[below of=A, yshift=0.25cm, xshift=-7mm, %anchor=north west, align=left, %text width=2.3*\c cm, fill=black!1, %draw=none, font=\normalsize %]{% %\begin{itemize} %\item Sei $w$ eine beliebige Ecktransversale von $C$; dann entliest man der Abbildung mit Hilfe des Sinussatzes %$\dfrac{\sin(\chi)}{\sin(\gamma_1)} = \dfrac{b}{m}$ und %$\dfrac{\sin(\overline{\chi})}{\sin(\gamma_1)} = \dfrac{a}{n}$.\\ % Für die supplementären Winkel bei $W$ ist $ %\sin(\overline{\chi}) = \sin(180^\circ -\chi) =\sin(\chi)$; also %$\sin(\chi)=\dfrac{b}{m}\cdot \sin(\gamma_1) %= \dfrac{a}{n}\cdot \sin(\gamma_2) =\sin(\overline{\chi})$. Damit erhält man die allgemeine Beziehung %$$\dfrac{b\sin(\gamma_1)}{a\sin(\gamma_2)} = \dfrac{m}{n}$$ %\item Für den Sonderfall $\gamma_1 =\gamma_2$ ist $w$ die Winkelhalbierende und man erhält %$$\dfrac{b}{a} = \dfrac{m}{n}$$ %\end{itemize} %}; %% Winkelhalbierende %\draw[] (C) --+ (-\Beta-0.5*\Gamma:\wC) coordinate[Punkt={below}{W_c}] (Wc); %\path[] (A) -- (Wc) node[midway, below]{$m_c$}; %\path[] (B) -- (Wc) node[midway, below]{$n_c$}; %\draw pic [draw, angle radius=3mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\gamma/2$", %] {angle =A--C--Wc}; %\draw pic [draw, angle radius=4mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\gamma/2$", %] {angle =Wc--C--B}; % %\draw[] (A) --+ (0.5*\Alpha:\wA) coordinate[Punkt={right}{W_a}] (Wa); %\path[] (B) -- (Wa) node[midway, above]{$m_a$}; %\path[] (C) -- (Wa) node[midway, above]{$n_a$}; %\draw pic [draw, angle radius=6mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\alpha/2$", %] {angle =Wa--A--C}; %\draw pic [draw, angle radius=5mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\alpha/2$", %] {angle =B--A--Wa}; % %\draw[] (B) --+ (180-0.5*\Beta:\wB) coordinate[Punkt={left}{W_b}] (Wb); %\path[] (A) -- (Wb) node[midway, left]{$m_b$}; %\path[] (C) -- (Wb) node[midway, left]{$n_b$}; %\draw pic [draw, angle radius=6mm, angle eccentricity=2, %% pic text={$\alpha$}, pic text options={}, %"$\beta/2$", %] {angle =Wb--B--A}; %\draw pic [draw, angle radius=8mm, angle eccentricity=1.7, %% pic text={$\alpha$}, pic text options={}, %"$\beta/2$", %] {angle =C--B--Wb}; % %%%% Punkte %\foreach \P in {Wa,Wb,Wc} \draw[fill=black!1] (\P) circle (1.75pt); \end{tikzpicture}$


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Wario
Aktiv Letzter Besuch: in der letzten Woche
Dabei seit: 01.05.2020
Mitteilungen: 947
  Beitrag No.1, vom Themenstarter, eingetragen 2022-07-05

$ % Seitenlängen \pgfmathsetmacro{\a}{6} % 4 \pgfmathsetmacro{\b}{4} % 3.5 \pgfmathsetmacro{\c}{7} % 6 \pgfmathsetmacro{\Alpha}{acos((\b^2+\c^2-\a^2)/(2*\b*\c))} \pgfmathsetmacro{\Beta}{acos((\a^2+\c^2-\b^2)/(2*\a*\c))} \pgfmathsetmacro{\Gamma}{acos((\a^2+\b^2-\c^2)/(2*\a*\b))} \pgfmathsetmacro{\wA}{2*\b*\c*cos(\Alpha/2)/(\b+\c)} \pgfmathsetmacro{\wB}{2*\a*\c*cos(\Beta/2)/(\a+\c)} \pgfmathsetmacro{\wC}{2*\a*\b*cos(\Gamma/2)/(\a+\b)} % Inkreis \pgfmathsetmacro{\s}{0.5*(\a+\b+\c)} \pgfmathsetmacro{\F}{sqrt(\s*(\s-\a)*(\s-\b)*(\s-\c))} \pgfmathsetmacro{\r}{\F/\s} \pgfmathsetmacro{\ai}{\a/(2*\s)} \pgfmathsetmacro{\bi}{\b/(2*\s)} \pgfmathsetmacro{\ci}{\c/(2*\s)} \pgfkeys{/tikz/savevalue/.code 2 args={\global\edef#1{#2}}} \begin{tikzpicture}[%scale=0.7, font=\footnotesize, background rectangle/.style={draw=none, fill=black!1, rounded corners}, show background rectangle, Punkt/.style 2 args={ label={[#1]:$#2$} }, Dreieck/.style={thick}, ] % Dreieckskonstruktion \coordinate[label=below:$A$] (A) at (0,0); \coordinate[label=below:$B$] (B) at (\c,0); \coordinate[label=$C$] (C) at (\Alpha:\b); \draw[local bounding box=dreieck] (A) -- (B) -- (C) --cycle; % Winkelhalbierende \draw[] (A) --+ (0.5*\Alpha:\wA) coordinate[label={[inner sep=1pt]45:$W_a$}] (Wa) node[pos=0.285, sloped, above, inner sep=1pt]{$w_\alpha$}; \path[] (B) -- (Wa) node[midway, right]{$a_B$}; \path[] (C) -- (Wa) node[near start, right]{$a_C$}; \draw[] (B) --+ (180-0.5*\Beta:\wB) coordinate[label={[inner sep=1pt]135:$W_b$}] (Wb) node[pos=0.3125, sloped, above, inner sep=1pt]{$w_\beta$}; \path[] (A) -- (Wb) node[midway, left]{$b_A$}; \path[] (C) -- (Wb) node[near start, left]{$b_C$}; \draw[] (C) --+ (-\Beta-0.5*\Gamma:\wC) coordinate[label=below:$W_c$] (Wc) node[pos=0.25, right, inner sep=1pt]{$w_\gamma$}; \path[] (A) -- (Wc) node[midway, below]{$c_A$}; \path[] (B) -- (Wc) node[midway, below]{$c_B$}; % Halbe Winkel \draw pic [draw, angle radius=10mm, angle eccentricity=0.8, % pic text={$\alpha$}, pic text options={}, "$\frac{\alpha}{2}$" ] {angle =B--A--Wa}; \draw pic [draw, angle radius=10mm, angle eccentricity=0.8, % pic text={$\alpha$}, pic text options={}, "$\frac{\alpha}{2}$" ] {angle =Wa--A--C}; \draw pic [draw, angle radius=15mm, angle eccentricity=0.9125, % pic text={$\alpha$}, pic text options={}, "$\frac{\beta}{2}$" ] {angle =C--B--Wb}; \draw pic [draw, angle radius=15mm, angle eccentricity=0.93, % pic text={$\alpha$}, pic text options={}, "$\frac{\beta}{2}$" ] {angle =Wb--B--A}; \draw pic [draw, angle radius=7mm, angle eccentricity=0.7, % pic text={$\alpha$}, pic text options={}, "$\frac{\gamma}{2}$" ] {angle =A--C--Wc}; \draw pic [draw, angle radius=7mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\frac{\gamma}{2}$" ] {angle =Wc--C--B}; % Inkreis \coordinate[label={[inner sep=1.0pt, yshift=1.5pt]75:$I$}] (I) at ($\ai*(A)+\bi*(B)+\ci*(C)$); %\draw[] (I) circle[radius=\r]; % Winkelhalbierenden-Dreieck \draw[local bounding box=dreieck] (Wa) -- (Wb) -- (Wc) --cycle; % Winkel am Winkelhalbierenden-Dreieck \draw pic [draw, angle radius=9mm, angle eccentricity=0.7, % pic text={$\alpha$}, pic text options={}, "$\overline\delta_a$" ] {angle =C--Wa--Wb}; \draw pic [draw, angle radius=9mm, angle eccentricity=0.8125, % pic text={$\alpha$}, pic text options={}, "$\overline\varepsilon_a$" ] {angle =Wb--Wa--A}; \draw pic [draw, angle radius=9mm, angle eccentricity=0.8, % pic text={$\alpha$}, pic text options={}, "$\varepsilon_a$" ] {angle =A--Wa--Wc}; \draw pic [draw, angle radius=9mm, angle eccentricity=0.7, % pic text={$\alpha$}, pic text options={}, "$\delta_a$" ] {angle =Wc--Wa--B}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\delta_b$" ] {angle =Wa--Wb--C}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\varepsilon_b$" ] {angle =B--Wb--Wa}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\overline\varepsilon_b$" ] {angle =Wc--Wb--B}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\overline\delta_b$" ] {angle =A--Wb--Wc}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\delta_c$" ] {angle =Wb--Wc--A}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\varepsilon_c$" ] {angle =C--Wc--Wb}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\overline\varepsilon_c$" ] {angle =Wa--Wc--C}; \draw pic [draw, angle radius=8mm, angle eccentricity=0.75, % pic text={$\alpha$}, pic text options={}, "$\overline\delta_c$" ] {angle =B--Wc--Wa}; %\path[] (A) -- (C) node[midway, left]{$b$}; %\path[] (B) -- (C) node[midway, right]{$a$}; %\draw pic [draw, angle radius=4mm, %angle eccentricity=1.8, % pic text={$\alpha$}, pic text options={}, %"$\chi$", %] {angle =C--W--A}; %\draw pic [draw, angle radius=5mm, %angle eccentricity=1.8, % pic text={$\alpha$}, pic text options={}, %"$\overline{\chi}$", %] {angle =B--W--C}; %% Punkte \foreach \P in {Wa, Wb, Wc, I} \draw[fill=black!1] (\P) circle (1.5pt); %% Annottion - Text %\node[below of=A, yshift=0.25cm, xshift=-7mm, %anchor=north west, align=left, %text width=2.3*\c cm, fill=black!1, %draw=none, font=\normalsize %]{% %\begin{itemize} %\item Sei $w$ eine beliebige Ecktransversale von $C$; dann entliest man der Abbildung mit Hilfe des Sinussatzes %$\dfrac{\sin(\chi)}{\sin(\gamma_1)} = \dfrac{b}{m}$ und %$\dfrac{\sin(\overline{\chi})}{\sin(\gamma_1)} = \dfrac{a}{n}$.\\ % Für die supplementären Winkel bei $W$ ist $ %\sin(\overline{\chi}) = \sin(180^\circ -\chi) =\sin(\chi)$; also %$\sin(\chi)=\dfrac{b}{m}\cdot \sin(\gamma_1) %= \dfrac{a}{n}\cdot \sin(\gamma_2) =\sin(\overline{\chi})$. Damit erhält man die allgemeine Beziehung %$$\dfrac{b\sin(\gamma_1)}{a\sin(\gamma_2)} = \dfrac{m}{n}$$ %\item Für den Sonderfall $\gamma_1 =\gamma_2$ ist $w$ die Winkelhalbierende und man erhält %$$\dfrac{b}{a} = \dfrac{m}{n}$$ %\end{itemize} %}; %% Winkelhalbierende %\draw[] (C) --+ (-\Beta-0.5*\Gamma:\wC) coordinate[Punkt={below}{W_c}] (Wc); %\path[] (A) -- (Wc) node[midway, below]{$m_c$}; %\path[] (B) -- (Wc) node[midway, below]{$n_c$}; %\draw pic [draw, angle radius=3mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\gamma/2$", %] {angle =A--C--Wc}; %\draw pic [draw, angle radius=4mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\gamma/2$", %] {angle =Wc--C--B}; % %\draw[] (A) --+ (0.5*\Alpha:\wA) coordinate[Punkt={right}{W_a}] (Wa); %\path[] (B) -- (Wa) node[midway, above]{$m_a$}; %\path[] (C) -- (Wa) node[midway, above]{$n_a$}; %\draw pic [draw, angle radius=6mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\alpha/2$", %] {angle =Wa--A--C}; %\draw pic [draw, angle radius=5mm, angle eccentricity=1.8, %% pic text={$\alpha$}, pic text options={}, %"$\alpha/2$", %] {angle =B--A--Wa}; % %\draw[] (B) --+ (180-0.5*\Beta:\wB) coordinate[Punkt={left}{W_b}] (Wb); %\path[] (A) -- (Wb) node[midway, left]{$m_b$}; %\path[] (C) -- (Wb) node[midway, left]{$n_b$}; %\draw pic [draw, angle radius=6mm, angle eccentricity=2, %% pic text={$\alpha$}, pic text options={}, %"$\beta/2$", %] {angle =Wb--B--A}; %\draw pic [draw, angle radius=8mm, angle eccentricity=1.7, %% pic text={$\alpha$}, pic text options={}, %"$\beta/2$", %] {angle =C--B--Wb}; % %%%% Punkte %\foreach \P in {Wa,Wb,Wc} \draw[fill=black!1] (\P) circle (1.75pt); \end{tikzpicture} $


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