| |
| Autor |
  Bezeichnung der Winkel am Dreieck aus den Seiten-Schnittpunkten mit den Winkelhalbierenden |
|
Wario
Aktiv 
Dabei seit: 01.05.2020
Mitteilungen: 1208
 |  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}$
|
 Profil
|
Wario
Aktiv 
Dabei seit: 01.05.2020
Mitteilungen: 1208
| 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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