I'm reading "Taxicab Geometry: Adventure in Non-Euclidean Geometry" by Eugene F Krause where he mentioned:
A geometry is called Euclidean if it satisfies the following thirteen properties:
- Given any two points there is exactly one line containing them.
- Every line contains at least two points, P contains at least 3 non-collinear points.
- The distance between two points is a non-negative number and is zero only if the two points coincide.
- The distance from A to B equals the distance from B to A.
- The distance function satisfies the triangle inequality property.
- Given any line, there exists a one-to-one and onto function that acts as a ruler for all pairs of points A, B on the line.
- Each line divides the space into two convex half-planes.
- The angle measurement function assigns to each angle a real number between 0 and 180.
- Given a ray ($\overrightarrow{AB}$) on the edge of a half-plane and given a real number r between 0 and 180, there is exactly one ray extending from A into the half-plane with the angle measure of r.
- The angle addition property.
- Two angles forming a straight angle sum to 180.
- The side-angle-side congruence holds for triangles.
- The parallel postulate: Given a point A not on a line l, there is exactly one line through A parallel to l.
May I ask, is there a name for such 13 properties?
