Document 3 - Basics (7-9)

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{hyperref}


\title{Document 3 - Basics}
\author{Thomas Boufikos}
\date{September 2021}

\begin{document}

\maketitle

\section{\LaTeX \, environments} \\ \\ \\
\subsection{Itemize}
\begin{center}
    \begin{itemize}
        \item Item 1
        \item[$\rightarrow$] Item 2
        \item Item 3
        \item[$\implies$] Item 4
    \end{itemize}
\end{center} \\ \\ \\

\subsection{Enumerate}
\begin{center}
    \begin{enumerate}
        \item Item 1
        \item[$\rightarrow$] Item 2
        \item Item 3
        \item[$\implies$] Item 4
    \end{enumerate}
\end{center} \\ \\ \\

\subsection{Theorems} \\ \\ \\

\newtheorem{thm}{Thomas Theorem} \\ \\
\begin{thm}
    This theorem is about Calculus! It says: $\frac{\sqrt{a}}{\sqrt{b}}
    = \sqrt{\frac{a}{b}}$.
\end{thm} \\ \\
\begin{thm}
    That theorem is also about Calculus! It says: \\
\[
    \int _a^b f(x) \, dx
    = F(x) \bigg\vert _a ^b
\]
\end{thm} \\ \\ \\

\newtheorem*{Pyth}{The Pythagorean Theorem}
\begin{Pyth}
    That theorem is about triangles with one angle $a = 90$ degrees: \\
    \[
        a^2 + b^2 = c^2
    \] \\
    ,where $a, b$ are the small angles and $c$ is the big angle.
\end{Pyth}
\\ \\ \\
\textbf{And now we will see new theorem environments with names:
"My Lemma", "Thomas Proposition" and "Nice proof"} \\ \\
\newtheorem{lem}{My Lemma}
\begin{lem}
    The first one.
\end{lem} \\ \\
\begin{lem}
    The second one.
\end{lem} \\ \\
\begin{lem}
    The third one.
\end{lem}
\\ \\
\newtheorem{prop}{Thomas Proposition}
\begin{prop}
    The first one.
\end{prop} \\ \\
\begin{prop}
    The second one.
\end{prop} \\ \\
\begin{prop}
    The third one.
\end{prop}
\\ \\
\begin{proof}
    There is nothing here!
\end{proof} \\ \\ \\ \\

\section{Random other things} \\ \\
\subsection{Hyphenation}
When someone says a big word like this: aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa there is a possibility to have bad notation. \\
But with hyphenation we can do the following one:aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa \-aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa.
\\
\subsection{Spaces} \\ \\
\textbf{Horizontal Space: } \\ \\
Instead of writing $ x = 1000 $ and $ y = 1000 $, we can do this: \\ \\
We can create bigger space: $ x = 1000 $ \hspace{2cm} and
\hspace{2cm} $ y = 1000 $. \\ \\ \\
\begin{center}
    \textbf{Test}
\end{center}
Name: \underline{\hspace{6cm}} \\
Age: \underline{\hspace{2cm}} \\
School: \underline{\hspace{8 cm}} \\ \\
Summation: \dotfill \\
\phantom{Phantom} Phantom \phantom{Phantom} Phantom \phantom{Phantom} Phantom \phantom{Phantom} Phantom \phantom{Phantom} Phantom \phantom{Phantom} Phantom \phantom{Phantom} Phantom \\
Again dotfill: \dotfill \\ \\ \\
\textbf{\textit{Black box with rule}} \\ \\
\begin{center}
    \rule{3cm}{3cm}
\end{center} \\ \\ \\
\section{Images} \\ \\ \\
\subsection{Image original}
\begin{center}
    \includegraphics{lena} \\ \\
\end{center}
\subsection{Image with new height = 6cm}
\begin{center}
    \includegraphics[height = 6cm]{lena} \\ \\
\end{center}
\subsection{Image with new width = 2cm}
\begin{center}
    \includegraphics[width = 2cm]{lena} \\ \\
\end{center}
\subsection{Image with new scale = 0.5}
\begin{center}
    \includegraphics[scale = 0.5]{lena} \\ \\
\end{center}
\subsection{Image (distorted) with new height = 6cm and width = 2cm}
\begin{center}
    \includegraphics[height = 6cm, width = 2cm]{lena} \\ \\
\end{center} \\ \\ \\ \\
\textbf{Internet hyperlink of the video about \LaTeX:}:
\begin{center}
    \href{https://www.youtube.com/watch?v=fCzF5gDy60g&ab_channel=AcademicLesson}{Click here}
\end{center}


\end{document}