Foundation of Geometry

Foundation of geometry is the study of geometric system that can be constructed in a special fields of mathematics. The word "foundation" can be judged from two different facts. First is the mathematical representation of the geometric nature of physical space that, we want represent mathematically entire of the universe. Second is the system standing by itself, which we can not study and property of a geometric figure in our space until we have established the space, have defined both the property and the figure, and have shown that the figure actually exist in the space. Before we began formal mathematics it is necessary to understand the nature of foundation properties. Some of the main foundation properties are given below.
(I) Existence Property
An existence property always states that there exists some figure with a given property. The property also states that how many of these figures are there in the system. When theexact number is one then the property is called a uniqueness theorem.
Example, for a given triangle, there exists a circle that passes through all three vertices of the triangle is the existence property of the circle for a given triangle and there is exactly one such circle which is the uniqueness property.
(II) Intersection property
The two sets, such as two line or a line a plane, do or do not intersect each other. If they intersect, then the intersection set either contains just one point or it contains the line. If a line and a plane do not intersect, then the intersection set is the empty set. If the two sets have at least one point in common are called incident sets and intersection properties are called incidence relation.
For example A intersection B