Even though synthetic geometry is superseded by analytic geometry, it is still interesting to study from a historical perspective as well as for understanding the nature of proofs.

Euclid's synthetic geometry was technically incomplete, however the axioms of Hilbert and Tarski have resolved these gaps.

Synthetic geometry has an advantage as being easy to axiomatically develop in contrrast to analytic geometry. Synthetic geometry requires a handful of axioms (20) and some definitions to exist on its own without any other mathematics other than logic.

Analytic geometry requires much more effort to define entirely, now commonly viewed as having set theory, topology, and linear algebra as its foundations. As a consequence as being connected to the 'mesh of mathematics', can draw from other fields of mathematics (notably mathematical analysis) for sophisticated results, however the lightweight nature of synthetic geometry still allows (and makes clearer!) some classic, purely geometric proofs that are elegant in their own way.


\chapter{Elements of Euclid's synthetic geometry}

Before doing any geometry, we need to understand the basic objects that we'll let exist in our space and , these ae sometimes refered to as elements of geometry (hence the name of Euclid's magnum opus).

We need geometric objects; shapes and constructs that exist within a space of Euclidean geometry.

We'll also define some geometric notions; measurements and quantities that Euclidean geometry considers.

Euclidean geometry is renown fo being the first 'axiomatic system' in mathematics; this makes Euclidean geometry interesting from a historical point of view and is usually among the first examples students are exposed to regarding proofs in mathematics. We will develop Euclidean geometry informally, however in the second part of this book we develop the foundations formally.

\section{Geometric objects}

\begin{definition}[Euclidean space (classic definition)]
\end{definition}

\begin{definition}[Point]
Uppercase Latin letters $A,B,C,\hdots$ are often used to represent points.
\end{definition}


\begin{definition}[Line]
\end{definition}

\begin{definition}[Plane]
\end{definition}
\section{Incidence}
\section{Betweenness}
\section{Congurence}

\section{Geometric notions}

\begin{definition}[Angle]
\end{definition}

\begin{definition}[Distance]
\end{definition}

- parallel lines
- perpendicular lines

- triangle
- scalene triangle
- isoceles triangle
- right angled triangle


- square
- rhombus
- rectangle
- parallelogram



- Pythagorean theoem
- isosceles triangle theorem (pons asinorum)
- Heron's fomula




\chapter{Compass \& straightedge constructions}
Compass equivalence theorem
Mohr-Mascheroni theorem
Poncelet-Steiner theorem

























\chapter{Axiomatization of Euclidean geometry}

Although ideas in mathematics have existed since prehistoric times, Thales of Miletus is often considered the first philosophe in the Greek tradition and the first mathematician in the west due to being the first to use of deductive reasoning fo mathematics, specifically geometry.

Euclid took this further with the formation of his axiomatic geometry and axiomatic arithmetic.

Nowadays, conventional mathematics is formalized by mathematical logic and ZFC. However, Euclid axiomatized geometry and number theory completely separately because he did not have the abstractions of set theory available to us today.

Although Euclid used some implicit assumptions in his easoning that weren't reflected by his axiomatic system, they were eventually patched and Euclidean geometry became the sole axiomatization of geometry.

More recently, other axiomatic systems of geometry (elliptic, hyperbolic, Riemannian etc.) have been developed to study geometry in different spaces where some Euclidean assumptions do not hold (in many cases it was the parallel postulate being violated), such as the surface of a sphere. Moreover, modern mathematics views geometry from the perspective of analytic geometry; the study of geometry through the notion of a Euclidean space from topology and linear algebra. Analytic geometry is preferrable since it retains all the properties of Euclidean geometry while also employing a coordinate scheme.

That said, Euclidean geomety remains a key milestone in mathematics, and its results are backwards compatible with analytic geomety through the Cantor-Dedekind theorem. Studying Euclidean geometry is not only the way that students learn basic geometry, but it also gives insight into the world of foundational mathematics and develops familiarity with formal proofs.

We will study Euclidean geometry from a modern perspective, taking a revized version of Euclid's axioms and postulates to avoid.




\section{Parallel postulate \& Playfair's axiom}

\section{Euclid's axioms}


\section{Modern Euclidean geometry}
Euclid's axioms are incomplete
Hilbert and Tarski's axioms are logically equivalent, however they have different 'logical flavours'
\subsection{Hilbert's axioms}
\subsection{Tarski's axioms}



\chapter{Non-Euclidean geometry}