如何用LaTeX Tikz畫幾何圖
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我這樣理解Tikz——
Tikz使用笛卡爾座標系和極座標表示一個點。
用\coordinate命令畫一個確定的點，但默認地，這個點像是像鉛筆在紙上點了一下，沒印出什麼來，直徑為零。如果要肉眼看到這個點，就用circle標出來。
用\node也可以畫一個點，但這個命令用途很廣，什麼圖案都可以是一個node。有時node占的空間大，不如coordinate精確，以我目前的理解，node是無法替代coordinate命令的。
本文的目標是給出一個可行的解決方案，是一個子集。
在coordinate確定點後，畫線用draw命令。也可以用path命令，但path命令默認是一個不可見的線段、三角形三條邊甚至一條曲線，path是圖論中的路徑的概念，一條通過多個節點的所經之路即path。我把draw當畫直線用，也可以用control參數畫曲線，但我沒有弄懂確切如何使用。我目前只當畫直線用。如此，我還不知道draw命令能否完全替代path。
在我有限的幾何圖繪畫中path用在畫角平分線，三角形的三條邊是一個path，從頂點畫一個圓，交兩條邊於兩個點，連接這兩個點，取中點，頂點連中點即角平分線。
角的示意符號使用pic參數，但如果不用，自己畫圓弧或者直角示意符號也是可以的。
關於函數如何繪畫，比如拋物線，我只是拷貝例子，還沒有自己使用。
我尚未整理所有的參考文檔，只在代碼註釋中。
過去我想著用Python畫圖，但初等數學能用LaTeX一種語言寫完，為什麼要多用一種計算機語言？這對非計算機專業的絕大多數人來說，不合適。所以我決定學習LaTeX畫幾何圖。本文是初稿。
LaTeX studio網站給出了很多複雜的圖形。我學習了其中全等三角形和相似三角形的符號。
我只貼PDF部分截圖。最後是LaTeX code，上傳文檔名為tikz_euclid_20250419.tex。
以下是LaTeX code。
% https://tikz.dev/tutorial-Euclid
% Typeset: XeLaTeX
\documentclass[12pt]{article}
\usepackage[margin=1in]{geometry}%設置頁邊距
\usepackage{ctex}%中文
\setCJKmainfont{Alibaba PuHuiTi} % 中文字體
\usepackage{fontspec} %英文
\setmainfont{Verdana} % 英文字體
\renewcommand{\baselinestretch}{1.5} % 設置1.5倍行距
\usepackage{hyperref}%支持鏈接可點擊
\usepackage[style=iso]{datetime2}%設置日期格式為YYYY--MM--DD
\usepackage{amsmath, amsfonts, amssymb}
\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{arrows}
\usetikzlibrary{datavisualization.formats.functions}
% 全等符號 \backcong和相似符號\backsim
%來源：厚德载物，『tikz绘制的全等与相似符号』，2022-07-25，https://www.latexstudio.net/index/details/index/mid/2816.html
%%%%%%%全等符号 定義開始%%%%%%%
\newcommand{ \usimeq}{\;
\begin{tikzpicture}[line width = 0.1 ex, line cap = round, shift={(1. 473ex, 1ex)}]
\draw (-0.1146ex, -0.2ex) arc (330:90:0. 4ex);
\draw (0.1146ex, 0.2ex) arc (150: -90:0.4ex);
\draw (-0.1146ex, -0.2ex) -- (0.1146ex, 0.2ex);
\draw (-0.9ex, -1.1ex) -- (0.9ex, -1.1ex);
\draw [yshift = 0.35ex] (-0.9ex, -1.1ex) -- (0.9ex, -1.1ex);
\end{tikzpicture}\;}
%%%%%%%全等符号 定義結束%%%%%%%
%%%%%%%相似符号 定義開始%%%%%%%
\newcommand{ \usim}{\;
\begin{tikzpicture}[line width = 0.1 ex, line cap = round, shift={(1. 473ex, 1ex)}]
\draw (-0.1ex, -0.25ex) arc (330:90:0.5ex);
\draw (0.1292ex, 0.25ex) arc (150: -90:0.5ex);
\draw (-0.1ex, -0.25ex) -- (0.1292ex, 0.25ex);
\end{tikzpicture}\;}
%%%%%%%相似符号 定義結束%%%%%%%
\parindent 0pt
%Example enviroment. Typeset the source code and print it in the PDF as well.
%https://tex.stackexchange.com/questions/240642/add-vertical-line-of-equation-x-2-and-shade-a-region-in-graph-by-pgfplots
\usepackage{fancyvrb-ex}
\title{\Huge\textbf{如何用LaTeX Tikz畫幾何圖}}
\author{}
\begin{document}
\maketitle
我這樣理解Tikz——
Tikz使用笛卡爾座標系和極座標表示一個點。
用\verb"\coordinate"命令畫一個確定的點，但默認地，這個點像是像鉛筆在紙上點了一下，沒印出什麼來，直徑為零。如果要肉眼看到這個點，就用circle標出來。
用\verb"\node"也可以畫一個點，但這個命令用途很廣，什麼圖案都可以是一個node。有時node占的空間大，不如coordinate精確，以我目前的理解，node是無法替代coordinate命令的。
本文的目標是給出一個可行的解決方案，是一個子集。
在coordinate確定點後，畫線用draw命令。也可以用path命令，但path命令默認是一個不可見的線段、三角形三條邊甚至一條曲線，path是圖論中的路徑的概念，一條通過多個節點的所經之路即path。我把draw當畫直線用，也可以用control參數畫曲線，但我沒有弄懂確切如何使用。我目前只當畫直線、圓弧用。如此，我還不知道draw命令能否完全替代path。
在我有限的幾何圖繪畫中path用在畫角平分線，三角形的三條邊是一個path，從頂點畫一個圓，交兩條邊於兩個點，連接這兩個點，取中點，頂點連中點即角平分線。
角的示意符號使用pic參數，但如果不用，自己畫圓弧或者直角示意符號也是可以的。
關於函數如何繪畫，比如拋物線，我只是拷貝例子，還沒有自己使用。
我甚至尚未整理所有的參考文檔，只在代碼註釋中。
過去我想著用Python畫圖，但初等數學能用LaTeX一種語言寫完，為什麼要多用一種計算機語言？這對非計算機專業的絕大多數人來說，不合適。所以我決定學習LaTeX畫幾何圖。本文是初稿。
\href{https://www.latexstudio.net}{LaTeX studio網站}給出了很多複雜的圖形。我學習了其中全等三角形和相似三角形的符號。
\section{如何在某一座標畫一個點}
\begin{Example}[frame=single, numbers=left]
\tikz \coordinate[shape=circle, fill=black, opacity=.5, %
inner sep=1pt] (A) at (0,0);
\end{Example}
% 座標表示 https://tikz.dev/tikz-design
\subsection{座標表示方法}
\begin{Example}[frame=single]
\tikz \draw (0,0) node[shape=circle, inner sep=1pt, %
fill=black!50,label=left:$A$] {} -- ++(1,0) -- (30:1cm);
%++(1,0)是相對座標，X軸右移1unit。(30:1cm)是極座標，30度角，半徑1cm。
\end{Example}
\section{如何畫一條線}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:\textcolor{orange}{$A$}] (A) at (0,0);
\coordinate [label=right:\textcolor{blue!80!black}{$B$}] (B) at (1,0.2);
\draw [red, thick] (A)--(B);
%
\coordinate [label=left:\textcolor{orange}{$A$}] (A) at (2,0);
\coordinate [label=right:\textcolor{blue!80!black}{$B$}] (B) at (3,0.2);
\path[draw=red, thick]
(A) -- (B);
%
\coordinate [label=left:\textcolor{orange}{$A$}] (A) at (4,0);
\coordinate [label=right:\textcolor{blue!80!black}{$B$}] (B) at (5,0.2);
\path
(A) edge [red, thick] (B);
\end{tikzpicture}
\end{Example}
%
\section{如何畫一個三角形}
\begin{Example}[frame=single]
\begin{tikzpicture}[roundpoint/.style={shape=circle, fill=black,%
opacity=.5, inner sep=1pt}]
% 第一種方法\coordinate定點，\fill circle畫小圓點
\coordinate [roundpoint, label=left:\textcolor{orange}{$A$}] %
(A) at (0,0);
\coordinate [roundpoint, label=right:\textcolor{blue!80!black}%
{$B$}](B) at (1,0.2);
\coordinate [roundpoint,label=above:\textcolor{red!80!black}{$C$}]%
(C) at (1,2);
\draw [thin,draw=black!50] (A) -- (B) -- (C) -- (A);
% \foreach \point in {A,B,C}
% \fill [black,opacity=.5] (\point) circle (2pt);
%
% 第2種方法 lable的方位用度數表示，X軸正向為0度。
\coordinate [label=180:\textcolor{orange}{$A$}] (A) at (2,0);
\coordinate [label=0:\textcolor{blue!80!black}{$B$}] (B) at (3,0.2);
\coordinate [label=90:\textcolor{red!80!black}{$C$}] (C) at (3,2);
\draw [thin,draw=black!50] (A) -- (B) -- (C) --(A);
\foreach \point in {A,B,C}
\fill [black!50] (\point) circle (1pt);
% 第3種方法 \node直接畫一個小圓點
\node [shape=circle, fill=black, opacity=.5, inner sep=1pt,%
label=left:$A$] (A) at (4,0) {};
\node [shape=circle, fill=black, opacity=.5, inner sep=1pt,%
label=right:$B$] (B) at (5,0.2) {};
\node [shape=circle, fill=black!50, inner sep=1pt,%
label=above:$C$] (C) at (5,2) {};
\draw[thin, black!50] (A) -- (B) -- (C) -- (A);
%
% 第4種方法 \draw (x,y) node直接畫一個小圓點, \path[draw]即\draw，
%但是\pah支持--cycle這個參數。
%
% 為了減少重複的語句，定義style。
% 定義style有兩種辦法：
% 1）\tikzset定義style可以在tikzpicture envionment之內，或之外。
% 2）在\begin{tikzpicure}[littlepoint/.style=……]可以定義style。
% 但另起\begin{tikzpicure}就換行了。如果可以發現，換行後，座標不起作用，
% 座標在Latex編譯中是相對位置。
%
\tikzset{littlepoint/.style={shape=circle, inner sep=1pt, fill=black!50}}
\path[draw=black!50] (6,0) node[littlepoint,label=left:$A$] {} -- (7,0.2)
node[littlepoint,label=right:$B$] {} -- (7,2)
node[littlepoint,label=above:$C$]{} -- cycle;
%
%綜合以上4種方法：用\coordinate定點畫點，或用\node定點畫點，\draw畫線，
%這樣比較清晰，我選第1或3種方法。
\end{tikzpicture}
\end{Example}
\section{如何畫一個矩形}
\begin{Example}[frame=single]
% https://tikz.dev/tikz-design
\begin{tikzpicture}
\path[draw=black!50] (5pt,0pt) -- (0pt,0pt) -- %
(0pt,5pt) -- cycle;
\path[draw=red] (1,0) rectangle (3,5);
\path[fill=black!50,draw=blue] (4,1) rectangle (5,4);
%\path[draw,clip] (1,1) rectangle (2,4);
%clip表示裁剪一個矩形框。
\draw[yellow] (6,6) rectangle (2,4);
\draw[green] (3.5,3) rectangle (6.5,7.5);
%上一句和這一句，可知：兩個座標高低沒有順序。
\end{tikzpicture}
\end{Example}
\section{如何畫一個圓}
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[black] (0,0) circle [radius=1cm];
\draw[blue] (0,0) circle [radius=10pt];
\draw[violet, fill=gray] (0,0) circle [radius=1ex];
\end{tikzpicture}
\end{Example}
\subsection{畫圓另一個方法}
\begin{Example}[frame=single]
\begin{tikzpicture}
\node[circle,draw,text=white,fill=lightgray] (c) at (0,0){Circle};
\end{tikzpicture}
\end{Example}
\subsection{如何畫一個半徑為座標軸上2unit的圓}
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[color=black, very thick] (1,0) circle (2);
\draw[color=black!30, very thick] (10,0) circle (2);
\end{tikzpicture}
\end{Example}
\subsection{如何畫一個半徑為1cm，顏色為橙色的圓}
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[color=black!60, very thick, fill=orange](1,0) circle [radius=1];
\draw[color=black!30, very thick] (2,0) circle (1);
\node [circle, fill=olive,inner sep=1cm, text=white,%
label=below:$X$] (X) at (10,0) {c1};
\end{tikzpicture}
\end{Example}
\subsection{如何畫圓心是A、半徑是AB的圓}
\usetikzlibrary {calc}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (3,3);
\coordinate [label=right:$B$] (B) at (1.25,0.25);
\draw (A) -- (B);
% \p1是一個vector，座標是\x1, \y1，用veclen計算radius
\draw (A) let
\p1 = ($ (B) - (A) $)
in
circle ({veclen(\x1,\y1)});
\end{tikzpicture}
\end{Example}
\section{如何畫一個橢圓}
\begin{Example}[frame=single]
\tikz \draw (0,0) ellipse [x radius=20pt, y radius=10pt];
\end{Example}
\section{如何畫X、Y座標軸}
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[->] (-1.5,0) -- (1.5,0) node[label=right:$x$]{};
\draw[->] (0,-1.5) -- (0,1.5) node[label=above:$y$]{};
\draw (0,0) circle [radius=1cm];
\end{tikzpicture}
\begin{tikzpicture}
\draw[->] (-1.5,0) -- (1.5,0) node[right]{$x$};
\draw[->] (0,-1.5) -- (0,1.5) node[above]{$y$};
\draw (0,0) circle [radius=1cm];
\end{tikzpicture}
\end{Example}
\subsection{如何畫X、Y、Z座標軸}
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[->] (0,0,0) -- (1.5,0,0) node[label=right:$x$]{};
\draw[->] (0,0,0) -- (0,1.5,0) node[label=above:$y$]{};
\draw[->] (0,0,0) -- (0,0,2.5) node[label=above:$z$]{};
\node[shape=circle, fill=gray, inner sep=1pt, %
label=right:$X$] (X) at (0.5,0.5,0.5) {};
\end{tikzpicture}
\end{Example}
\section{如何畫一段圓弧}
\begin{Example}[frame=single]
\begin{tikzpicture}[scale=3]
\draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
\draw[->] (-1.5,0) -- (1.5,0) node[label=right:$x$]{};
\draw[->] (0,-1.5) -- (0,1.5) node[label=above:$y$]{};
\draw (0,0) circle[radius=1cm];
\draw[fill=gray, opacity=0.5] (-0.5,0)--(0.5,0) %
arc [start angle=0, end angle=135, radius=1cm] -- (-0.5,0) ;
\end{tikzpicture}
\end{Example}
\section{如何畫拋物線}
\begin{Example}[frame=single]
\begin{tikzpicture}[scale=3]
\draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
\draw[->] (-1.5,0) -- (1.5,0) node[label=right:$x$]{};
\draw[->] (0,-1.5) -- (0,1.5) node[label=above:$y$]{};
\draw (0,0) parabola (1,1);
\draw (-0.5,-0.5) parabola bend (1,0.5) (1.4,0.4);
\end{tikzpicture}
\end{Example}
%https://tikz.dev/dv-main
\begin{Example}[frame=single]
\usetikzlibrary {datavisualization.formats.functions}
\begin{tikzpicture}
\datavisualization [school book axes, visualize as smooth line]
data [format=function] {
var x : interval [-1.5:1.5] samples 10;
func y = \value x*\value x;
};
\end{tikzpicture}
\end{Example}
\subsection{更多和拋物線有關的例子}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{patterns}
\begin{Example}[frame=single]
\pgfplotsset{compat=1.12}
\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
xmin=-1, xmax=3,
ymin=-1, ymax=3,
axis lines=middle,
]
\addplot[samples=100, domain=0:3, name path=A] {3/(2*x+1)};
\addplot[samples=50, domain=0:3,name path=B] {3*x-2};
\path[name path=xaxis] (\pgfkeysvalueof{/pgfplots/xmin}, 0)%
-- (\pgfkeysvalueof{/pgfplots/xmax},0);
\addplot[gray, pattern=north west lines] fill %
between[of=B and xaxis, soft clip={domain=0.66666:1}];
\addplot[gray, pattern=north west lines] fill %
between[of=A and xaxis, soft clip={domain=1:2}];
\addplot +[mark=none] coordinates {(2, -1) (2, 3)};
\end{axis}
\end{tikzpicture}
\end{figure}
\end{Example}
這個例子摘錄，但我不理解。
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[gray,very thin] (-1.9,-1.9) grid (2.9,3.9)
[step=0.25cm] (-1,-1) grid (1,1);
\draw[blue] (1,-2.1) -- (1,4.1); % asymptote
\draw[->] (-2,0) -- (3,0) node[right] {$x(t)$};
\draw[->] (0,-2) -- (0,4) node[above] {$y(t)$};
\foreach \pos in {-1,2}
\draw[shift={(\pos,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {$\pos$};
\foreach \pos in {-1,1,2,3}
\draw[shift={(0,\pos)}] (2pt,0pt) -- (-2pt,0pt) %
node[left] {$\pos$};
\fill (0,0) circle (0.064cm);
\draw[thick,parametric,domain=0.01:5,samples=200]
% The plot is reparameterised such that there are more samples
% near the center.
plot[id=asymptotic-example] function{(t)*sin(1/(t)),%
(t)*cos(1/(t))}
node[above right] %
{$\bigl(x(t),y(t)\bigr) = (t\sin \frac{1}{t}, t\cos \frac{1}{t})$};
\fill[red] (0.63662,0) circle (2pt)
node [below right,fill=white,yshift=-4pt] %
{$(\frac{2}{\pi},0)$};
\end{tikzpicture}
\end{Example}
這個例子來自官方tikz manual文檔，但我不理解。
\subsection{如何畫正弦、餘弦曲線}
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[step=.5cm,gray,very thin] (-2,-2) grid (4,4);
\draw[->] (-2,0) -- (4,0) node[label=right:$x$]{};
\draw[->] (0,-2) -- (0,4) node[label=above:$y$]{};
\draw (0,0) sin (1,1) cos (2,0) sin (3,-1) cos (4,0) ;
\draw[blue] (-2,0) cos (-1,-1) ;
\draw[violet,thick] (-1,-1) cos (0,0);
\draw[red,thick] (-2,0) sin (-1,-1) ;
\draw[yellow] (-1,-1) sin (0,0);
\end{tikzpicture}
\end{Example}
%https://tex.stackexchange.com/questions/356488/add-a-vertical-line-to-tikzpicture-plot
%\usetikzlibrary{arrows}
%\usetikzlibrary{datavisualization.formats.functions}
%\usepackage{pgfplots}
\begin{Example}[frame=single]
\begin{tikzpicture}
\datavisualization [school book axes,
visualize as smooth line/.list={one,two,three},
y axis={label={$u(t)$}},
x axis={label={$t$}},
one={style={black}},
two={style={red}},
three={style={orange}}]
data [set=one,format=function] {
var x : interval [-3:3];
func y = sin((\value x) r );
}
data [set=two,format=function] {
var x : interval [-3:3];
func y = sin((1.5 - \value x) r);
};
\draw [dashed] (1.5,-1) -- (1.5,1);
\end{tikzpicture}
\end{Example}
%https://tikz.dev/dv-main
\usetikzlibrary {datavisualization.formats.functions}
\begin{Example}[frame=single]
\begin{tikzpicture}[baseline]
\datavisualization [ scientific axes=clean,
y axis=grid,
visualize as smooth line/.list={sin,cos,tan},
style sheet=strong colors,
style sheet=vary dashing,
sin={label in legend={text=$\sin x$}},
cos={label in legend={text=$\cos x$}},
tan={label in legend={text=$\tan x$}},
data/format=function ]
data [set=sin] {
var x : interval [-0.5*pi:4];
func y = sin(\value x r);
}
data [set=cos] {
var x : interval [-0.5*pi:4];
func y = cos(\value x r);
}
data [set=tan] {
var x : interval [-0.3*pi:.3*pi];
func y = tan(\value x r);
};
\end{tikzpicture}
\end{Example}
\section{如何畫角}
一根直線可以起名，這樣直線的交點可以用intersection定點。
\usetikzlibrary {intersections}
\begin{Example}[frame=single]
\begin{tikzpicture}[scale=3]
\clip (-0.1,-0.2) rectangle (1.1,1.51);
\draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
\draw[->] (-1.5,0) -- (1.5,0);
\draw[->] (0,-1.5) -- (0,1.5);
\draw (0,0) circle [radius=1cm];
\filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
arc [start angle=0, end angle=30, radius=3mm] -- cycle;
\draw[red,very thick] (30:1cm) -- +(0,-0.5);
\draw[blue,very thick] (30:1cm) ++(0,-0.5) -- (0,0);
\path [name path=upward line,draw] (1,0) -- (1,1);
\path [name path=sloped line,draw] (0,0) -- (30:1.5cm);
\draw [name intersections={of=upward line and sloped line, by=x}]
[very thick,orange] (1,0) -- (x);
\end{tikzpicture}
\end{Example}
\subsection{如何畫角的指示，以及如何添加註釋文字}
直線用直線上兩點命名，直線的交點intersection cs用first line、second line。
\usetikzlibrary{angles, calc, quotes}
\begin{Example}[frame=single]
\begin{tikzpicture}[angle radius=.75cm]
%請注意觀察A與B的差別
\node[shape=circle,inner sep=2pt, fill=black!50,label=above:$A$]%
(A) at (-2,0) [red] {};
\node (B) at ( 3,.5) %
[shape=circle,inner sep=1pt, fill=black!50, text=black] {$B$};
\node (C) at (-2,2) [blue,left] {$C$};
\node (D) at ( 3,2.5) [blue,right] {$D$};
\node (E) at (60:-5mm) [below] {$E$};
\node (F) at (60:3.5cm) [above] {$F$};
\node[shape=circle, inner sep=1pt, fill=blue, label=-45:\tiny $x$]%
(x) at (intersection cs:first line={(A)--(B)}, %
second line={(E)--(F)}) {};
\node[shape=circle, inner sep=1pt, fill=blue, label=20:\tiny $y$]%
(y) at (intersection cs:first line={(C)--(D)}, %
second line={(E)--(F)}){};
\path
(A) edge [red, thick] (B)
(C) edge [blue, thick] (D)
(E) edge [thick] (F)
pic ["$\alpha$", draw, fill=yellow] {angle = F--x--A}
pic ["$\beta$", draw, fill=green!30] {angle = B--x--F}
pic ["$\gamma$", draw, fill=yellow] {angle = E--y--D}
pic ["$\delta$", draw, fill=green!30] {angle = C--y--E};
\node at ($ (D)!.5!(B) $) [right=1cm,text width=6cm,%
rounded corners, fill=red!20, inner sep=1ex]
{
When we assume that $\color{red}AB$ and $\color{blue}CD$ %
are parallel, i.\,e., ${\color{red}AB} \mathbin{\|} %
\color{blue}CD$,
then $\alpha = \gamma$ and $\beta = \delta$.
};
\end{tikzpicture}
\end{Example}
\usetikzlibrary {angles}
\begin{Example}[frame=single]
\tikz \draw (2,0) coordinate (A) -- (0,0) coordinate (B)
-- (-1,-1) coordinate (C)
pic [fill=black!50] {angle = A--B--C}
pic [draw,->,red,thick,angle radius=1cm] {angle = C--B--A};
\end{Example}
%Questioner: marmistrz, Answerer: rpapa, ``Create angle bisectors in TikZ'', \url{https://tex.stackexchange.com/questions/233246/create-angle-bisectors-in-tikz}
\begin{Example}[frame=single]
\usetikzlibrary {angles}
\tikz
\draw (0,0,0) coordinate (O)
(1,0,0) coordinate (A) -- (O)
(0,0,1) coordinate (B) -- (O)
(0,1,0) coordinate (C) -- (O)
pic [fill=gray,angle radius=4mm] {right angle = A--O--B}
pic [draw,red,thick,angle eccentricity=.5,pic text=.]
{right angle = A--O--C};
\end{Example}
\subsection{如何摘取一部分圖，放大}
\begin{Example}[frame=single]
\begin{tikzpicture}[scale=3]
\clip (-0.1,-0.2) rectangle (1.1,1.51);
\draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
\filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
arc [start angle=0, end angle=30, radius=3mm] -- cycle;
\draw[->] (-1.5,0) -- (1.5,0);
\draw[->] (0,-1.5) -- (0,1.5);
\draw (0,0) circle [radius=1cm];
\foreach \x in {-1cm,-0.5cm,1cm}
\draw (\x,-1pt) -- (\x,1pt);
\foreach \y in {-1cm,-0.5cm,0.5cm,1cm}
\draw (-1pt,\y) -- (1pt,\y);
\end{tikzpicture}
\end{Example}
\subsection{如何畫一個矩形}
座標系左下角為(0,0)
\begin{Example}[frame=single]
\begin{tikzpicture}
\draw[color=black, thick, fill=blue!80!white] (5,0) rectangle (10,1);
\fill[blue!40!white] (0,0) rectangle (4,4);
\fill[blue!60!red!90!green] (10,1) rectangle (12,4);
\end{tikzpicture}
\end{Example}
\section{如何畫直線上某一位置的點}
\subsection{如何畫AB中點}
\begin{Example}[frame=single]
\usetikzlibrary {calc}
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=right:$B$] (B) at (1.25,0.25);
\draw (A) -- (B);
\node [shape=circle, fill=red,inner sep=1pt,label=below:$X$]%
(X) at ($ (A)!.5!(B) $) {};
\end{tikzpicture}
\end{Example}
\subsection{如何畫底邊中線}
%https://tikz.dev/math-parsing
\begin{Example}[frame=single]
\usetikzlibrary {calc,quotes}
\tikz{
\coordinate["$A$"] (A) at (2,2);
\coordinate["$B$" below] (B) at (0,0);
\coordinate["$C$" below] (C) at (3,0);
\draw (A) -- (B) -- (C) -- cycle;
\path
let \p1 =($(A)-(B)$), \p2 =($(A)-(C)$),
\n1 = {veclen(\x1,\y1)}, \n2 = {veclen(\x2,\y2)}
in coordinate ["$D$" below] (D) at ($ (B)!0.5!(C) $);
\draw (A) -- (D);
}
\end{Example}
\subsection{如何畫出線段上某一比例的點}
\begin{Example}[frame=single]
\usetikzlibrary {calc,quotes}
\tikz{
\coordinate["$A$"] (A) at (2,2);
\coordinate["$B$" below] (B) at (0,0);
\coordinate["$C$" below] (C) at (3,0);
\draw (A) -- (B) -- (C) -- cycle;
\path
let \p1 =($(A)-(B)$), \p2 =($(A)-(C)$),
\n1 = {veclen(\x1,\y1)}, \n2 = {veclen(\x2,\y2)}
in coordinate ["$D$" below] (D) at%
($ (B)!scalar(\n1/(\n1+\n2))!(C) $);
\draw (A) -- (D);
}
\end{Example}
\subsection{如何畫AB延長線上的距離A點3.5倍距離的一點C}
\begin{Example}[frame=single]
\usetikzlibrary {calc}
\begin{tikzpicture}[redpoint/.style={shape=circle, %
fill=red,inner sep=1pt}]
\coordinate [redpoint, label=below:$A$] (A) at (0,0);
\coordinate [redpoint, label=below:$B$] (B) at (1.25,0.25);
\draw (A) -- (B) ;
\draw (A) -- ($ (A) ! 3.5 ! (B) $) coordinate %
[redpoint,label=below:$C$] (C) ;
\end{tikzpicture}
\end{Example}
\subsection{如何畫一個角，45度，一條邊是另一個條邊的兩倍長}
\usetikzlibrary {calc,through}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=right:$B$] (B) at (0.75,0.25);
\draw (A) -- (B);
\coordinate [label=above:$D$] (D) at
($ (A) ! 2 ! 45:(B) $) {};
\draw (A) --(D) --(B);
\end{tikzpicture}
\end{Example}
\subsection{如何畫一個正三角形}
\usetikzlibrary {calc,through}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=right:$B$] (B) at (0.75,0.25);
\draw (A) -- (B);
\coordinate [label=above:$D$] (D) at
($ (A) ! 0.5 ! (B) ! sin(60)*2 ! 90:(B) $) {};
\draw (A) --(D)--(B);
\end{tikzpicture}
\end{Example}
\subsection{一個複雜幾何圖例子}
\usetikzlibrary {calc,intersections,through}
\begin{Example}[frame=single]
\begin{tikzpicture}[thick,help lines/.style={thin,draw=black!50}]
\def\A{\textcolor{orange}{$A$}} \def\B{\textcolor{input}{$B$}}
\def\C{\textcolor{input}{$C$}} \def\D{$D$}
\def\E{$E$} \def\F{$F$}
\def\G{$G$} \def\H{$H$}
\def\K{$K$} \def\L{\textcolor{output}{$L$}}
\colorlet{input}{blue!80!black} \colorlet{output}{red!70!black}
\coordinate [label=left:\A] (A) at ($ (0,0) + .1*(rand,rand) $);
\coordinate [label=right:\B] (B) at ($ (1,0.2) + .1*(rand,rand) $);
\coordinate [label=above:\C] (C) at ($ (1,2) + .1*(rand,rand) $);
\draw [input] (B) -- (C);
\draw [help lines] (A) -- (B);
\coordinate [label=above:\D] (D) at ($ (A)!.5!(B) ! {sin(60)*2} ! 90:(B) $);
\draw [help lines] (D) -- ($ (D)!3.75!(A) $) coordinate [label=-135:\E] (E);
\draw [help lines] (D) -- ($ (D)!3.75!(B) $) coordinate [label=-45:\F] (F);
\node (H) at (B) [name path=H,help lines,circle through=(C),draw,label=135:\H] {};
\path [name path=B--F] (B) -- (F);
\path [name intersections={of=H and B--F,%
by={[label=right:\G]G}}];
\node (K) at (D) [name path=K,help lines,circle through=(G),draw,label=135:\K] {};
\path [name path=A--E] (A) -- (E);
\path [name intersections={of=K and A--E,%
by={[label=below:\L]L}}];
\draw [output] (A) -- (L);
\foreach \point in {A,B,C,D,G,L}
\fill [black,opacity=.5] (\point) circle (2pt);
% \node ...
\end{tikzpicture}
\end{Example}
\section{如何畫輔助線虛線}
畫一個鈍角三角形，40度，一條邊是另一個條邊的3倍長。
方法1
\usetikzlibrary {calc,through}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=below:$B$] (B) at (1,0);
\draw (A) -- (B);
\coordinate [label=above:$C$] (C) at
($ (A) ! 3 ! 40:(B) $) {};
\draw (A) --(C) --(B);
\draw[dashed] (B) -- ($(A) ! 3!(B)$) coordinate (D);
\coordinate (E) at ($(C)!1!50:(A)$);
\coordinate[label=below:$F$] (F) at%
(intersection cs:first line={(B)--(D)}, %
second line={(C)--(E)});
\draw[dashed] (C) -- (F);
\draw (F)++(0,3pt) -- ++(3pt,0pt) -- ++(0,-3pt);
\end{tikzpicture}
\end{Example}
方法2
\usetikzlibrary {calc,through}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=below:$B$] (B) at (1,0);
\draw (A) -- (B);
\coordinate [label=above:$C$] (C) at
($ (A) ! 3 ! 40:(B) $) {};
\draw (A) --(C);
\draw (B) -- (C);
% 畫AB的延長線，line1
\draw[dashed] (B) -- ($(A) ! 3!(B)$) coordinate (Y);
% 畫C向下垂直的線，line2
\path (C) -- ++(-90:2) coordinate (X);
\coordinate[label=below:$F$] (F) at%
(intersection cs:first line={(C)--(X)}, %
second line={(B)--(Y)});
\draw[dashed] (C) -- (F);
\draw (F)++(0,3pt) -- ++(3pt,0pt) -- ++(0,-3pt);
\end{tikzpicture}
\end{Example}
方法3
\usetikzlibrary {calc,through}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=below:$B$] (B) at (1,0);
\draw (A) -- (B);
\coordinate [label=above:$C$] (C) at (40:3);
\draw (A) --(C);
\draw (B) -- (C);
% 畫AB的延長線，line1
\draw[dashed, name path=line1] (B) -- %
($(A) ! 3!(B)$) coordinate (Y);
% 畫C向下垂直的線，line2
\path[name path=line2] (C) -- ++(-90:2) coordinate (X);
\draw [name intersections={of=line1 and line2, by=F}]%
[dashed] (C) -- (F) node[label=below:$F$]{};
\draw (F)++(0,3pt) -- ++(3pt,0pt) -- ++(0,-3pt);
\end{tikzpicture}
\end{Example}
方法4
\usetikzlibrary {calc,through}
\begin{Example}[frame=single]
\begin{tikzpicture}
\coordinate [label=left:$A$] (A) at (0,0);
\coordinate [label=below:$B$] (B) at (1,0);
\coordinate [label=above:$C$] (C) at (40:3);
\draw (A) -- (B) --(C) --cycle;
%C在AB上的投影點
\coordinate [label=below:$F$] (F) at ($(A)!(C)!(B)$);
% 畫AB的延長線，line1
\draw[dashed, name path=line1] (B) -- ($(A) ! 3!(B)$) %
coordinate (Y);
\draw[dashed, name path=line2] (C) --(F);
\draw (F)++(0,3pt) -- ++(3pt,0pt) -- ++(0,-3pt);
\end{tikzpicture}
\end{Example}
結論：方法4最好，Tikz可以在一條線段上畫出某一點的投影點，即垂線的垂足。
\section{如何畫角平分線}
\begin{Example}[frame=single]
\usetikzlibrary{positioning,calc,intersections}
\begin{tikzpicture}
\coordinate[label=below left:$C$](C) at (-2,0);
\coordinate[label=below right:$A$](A) at (8,0);
\coordinate[label=above left:$B$] (B) at (0,7);
\coordinate[label=above right:$D$](D) at ($(A)!(C)!(B)$);
% first name the path
\draw[name path=t1] (A) -- (B) -- (C) -- cycle;
\draw[name path=CD] (C) -- (D);
% draw circle
\path[name path=c1] (B) circle (2);
% name intersection
\path [name intersections={of = c1 and t1, by={x,y}}];
% draw line beetwen intersections
\path (x)--(y);
% draw the bissector
\path[name path=bissec] (B) -- ($(x)!0.5!(y)$) coordinate[pos=3](e)--(e);
\path [name intersections={of = CD and bissec, by=F}];
\draw[thick,red] (B) --(F);
\end{tikzpicture}
\end{Example}
\subsection{角平分線舉例}
已知$AD$、$AE$分別是$\triangle ABC$的角$A$的內、外角平分線，點$D$在邊$BC$上，點$E$在邊$BC$上的延長線上。求證：$\displaystyle \frac{1}{BE}+\frac{1}{CE}=\frac{2}{DE}$。
\usetikzlibrary{positioning,calc,intersections}
\begin{Example}[frame=single]
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\coordinate[label=below:$B$] (B) at (0,0);
\coordinate[label=45:$C$] (C) at (3,0);
\coordinate[label=above:$A$] (A) at (35:5);
\draw[name path=t1] (B)--(C)--(A)--cycle;
% start drawing 1st angle bisection
\path[name path=c1] (A) circle (1);
% name intersections
\path[name intersections = {of = t1 and c1, by={x, y}}];
% draw angle bisection
\coordinate (z) at ($(x)!.5!(y)$) ;
\coordinate (u) at ($(A) !3.4! (z)$);
% Intersection of angle bisection and the edge
\coordinate[label=below:$D$] (D) at (intersection cs:first line={(A)--(u)}, second line={(B)--(C)});
\draw (A)--(D);
\draw pic[draw, angle radius = 0.5cm] {angle = B--A--D};
\draw pic[draw, angle radius = 0.6cm] {angle = D--A--C};
%start drawing 2nd angle bisection
\coordinate[label=above:$P$] (P) at ($(B)!1.6!(A)$);
\draw[dashed] (A)--(P);
\coordinate (v) at ($(B)!3.1!(C)$);
\draw[name path=e1] (C)--(v);
\path[name path=b2] (A) -- ($(A)!3!90:(D)$);
\path[name intersections = {of = e1 and b2, by = w}];
\coordinate[label=below:$E$] (E) at (w);
\draw (A) -- (E);
\draw pic[draw,double, angle radius = 0.5cm] {angle = C--A--E};
\draw pic[draw,double, angle radius = 0.6cm] {angle = E--A--P};
% Help lines
\coordinate[label=above:$F$] (F) at ($(A)!(D)!(B)$);
\draw[dashed] (D) -- (F);
\draw pic[draw, angle radius = 0.2cm] {right angle = D--F--A};
\coordinate[label=below:$G$] (G) at ($(A)!(D)!(C)$);
\draw[dashed] (D) -- (G) -- (C);
\draw pic[draw, angle radius = 0.2cm] {right angle = D--G--A};
\coordinate[label=below:$H$] (H) at ($(A)!(E)!(C)$);
\draw[dashed] (E) -- (H) ;
\draw pic[draw, angle radius = 0.2cm] {right angle = E--H--A};
\coordinate[label=above:$I$] (I) at ($(A)!(E)!(P)$);
\draw[dashed] (E) -- (I) ;
\draw pic[draw, angle radius = 0.2cm] {right angle = E--I--A};
\end{tikzpicture}
\end{figure}
\end{Example}
證明：
思路：$DE = BE - BD$，$DE = DC + CE$，所以$\displaystyle \frac{1}{BE}+\frac{1}{CE}=\frac{2}{DE}$可化為$\displaystyle \frac{BD}{BE} = \frac{DC}{CE}$（目標公式）。角平分線的點畫垂線可得到全等直角三角形。\
添加輔助線，通過$D$向$AB$、$AC$畫垂線，分別為$DF$、$DG$，$DF = DG$；通過$E$向$AC$、$AP$畫垂線，分別為$EH$、$EI$，$EH = EI$。因為角平分線，可知\
$\because \triangle BDF\usim \triangle BEI$ \
$\displaystyle \therefore \frac{BD}{BE} = \frac{DF}{EI}$\
$\because \triangle DGC\usim \triangle BHC$ \
$\displaystyle \therefore \frac{DC}{CE} = \frac{DG}{EH}$\
故， $\displaystyle \frac{BD}{BE} = \frac{DC}{CE}$，得到目標公式。證畢。
\begin{thebibliography}{99}
\bibitem{hdzw}
厚德载物，``tikz绘制的全等与相似符号''，2022-07-25，\url{https://www.latexstudio.net/index/details/index/mid/2816.html}
\end{thebibliography}
\end{document}
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