\part{Fundamentals}


Geometry is the study of shapes, space, and their properties; it is, along with arithmetic, the oldest branch of mathematics.


The Greek mathematician attempted to define a 'system of geometry', in which one only has a space with points and lines as objects that follow certain rules. From these basic rules (called axioms), one uses them to do geometry; this is called synthetic geometry. In modern times, mathematics has evolved substantially; we have developed a more effective way of doing geometry that is connected with the rest of mathematics by the use of 'coordinates' (analytic geometry). 


We will look at geometry from both a classical 'synthetic' point of view as well as from a modern 'analytic' perspective throughout this book.


\chapter{Shapes}


\section{Elements}
In orer to create shapes, we will introduce some basic tools that we will use to describe objects.

A point is theoretical particle that simply determines a position in  space; it has no other properties.

A line segment is a direct connection between 2 points. the length of a line segment is the distance between the points the line segment connects.

Parallel lines
Perpendicular lines



\section{Polygons}

\section{Perimeter, area, volume, and angles}
\section{Triangles}

\section{Circle}

All of the shapes we have looked at so far are polygons (i.e made of lines), however there is one curved shape with which we can still do much basic geometry on; the circle!

Originally, mathematicians viewed circles in the following way.
\begin{definition}[Circle]
A \emph{circle} is the shape made by all the points of a fixed distance (called a \emph{radius}) from a point (\emph{center of origin}). The distance from one end of the circle to the opposite end is the \emph{diameter}.
\end{definition}

- diameter
- radius
- chord
- arc
- segment
- tangent
- secant
Here is a picture demonstrating the circle.

INSERT IMAGE HERE.


\[C=2\pi r\]
\[A=\pi r^2\]


\begin{theorem}[Interior angle sum]
\[180(n-2)\]
\end{theorem}

\chapter{Trigonometry}
\section{Trigonometric functions}
\subsection{Congruence between right angle triangles}
right angle triangle
opposite side
adjacent side
hypotenuse

right angle triangles
\subsection{Trigonometric functions}
- sine
- cosine
- tangent

\section{Inverse trigonometric functions}

Just ike how the square root is the 'inverse' of squaring since it calculates what we'd need to square to get a desired number, there.
These inverse trigonometric functions calculate what angle we would need so that we obtain a desired ratio of sides of a right angled triangle.
- arcsine
- arccosine
- arctangent





\section{Theorems of trigonometry}

The pythagoreal theorem applied to a right angle triangle of hypotenuse 1 gives us the following.
\begin{theorem}
\[\sin^2 (\theta) + \cos^2 (\theta) = 1\]
\end{theorem}

\begin{theorem}[Sine rule]
\[\frac{\sin (\alpha)}{a} = \frac{\sin (\beta)}{b} = \frac{\sin (\gamma)}{c}\]
\end{theorem}

\begin{theorem}[Cosine rule]
\[c^2 = a^2 +b^2 -2ab \cos (\gamma)\]
\[\cos (\gamma) =  \frac{a^2 +b^2 -c^2}{2ab} \]
\end{theorem}



\begin{definition}[Degrees]
A full rotation is defined as $360$ degrees.
\end{definition}

\begin{definition}[Radians]
A full rotation is defined as $2\pi$ radians.
\[\mathrm{rad}=\frac{\pi}{180} \mathrm{deg}\]
\end{definition}
The idea is that
-
 
\subsection{Further properties of trigonometric functions}

\[\sin (\theta) = \cos (\pi - \theta)\]




\chapter{Coordinate system}

\section{Cartesian coordinates}
\section{Vectors}
\section{Algebraic curves}
\subsection{Line equation}
\subsection{Circle equation}
\subsection{Ellipse equation}
\subsection{Points of intersection}
Calculus is extremely powerful in the domain of coordinate geometry and algebraic curves. Algebraic curves are still a topic of research as 'algebraic geometry'.