Synthetic geometry differs from other branches of geometry because it focuses on pure geometrical contents and draws conclusions through the use of axioms, logical arguments and theorems. Synthetic geometry analyzes the incidence relationship between objects such as lines and planes. These objects and their relationship of incidence are known as the primitives of synthetic geometry.
Axiomatic geometry is sometimes used interchangeably with the term synthetic geometry. In mathematics, the concept of an axiom is critical to the study of Euclidean geometry. Axioms are statements which are used to describe and prove the primitives of synthetic geometry. Essentially, these are just basic statements which have not yet been accepted or proven. For example, the statement “for any two distinct points, there is only one line which passes through both of these points”, is an axiom.
In conjunction with axioms, theorems are used in synthetic geometry. Theorems differ from axioms because they represent proven statements. Theorems can be proven by axioms which are thought to be accepted or by other theorems which exist. In order to formulate conclusions regarding mathematical problems and conduct geometric computations, theorems are utilized in synthetic geometry. Furthermore, logical arguments are used as a mode of proof through persuasion.
Evidently, synthetic geometry is one of the many distinct branches of geometry, just like analytic and algebraic geometry, which is used to understand geometrical structures. Thus, understanding the proper use of axioms, logical arguments and theorems is essential to this field of thought.