Download Axioms of Fano`s geometry Undefined Terms: point, line 1. There is

Axioms of Fano’s geometry
Undefined Terms: point, line
1. There is at least one line.
2. There are exactly three points on every line.
3. Not all points are on the same line.
4. There is exactly one line that is on any two distinct points.
5. There is at least one point that is on any two lines.
Incidence Axioms
Undefined Terms: point, line
1. For each two distinct points, there exists a unique line on both of them.
2. For every line, there exist at least two distinct points on it.
3. There exist at least three distinct points.
4. Not all points lie on the same line.
Euclid’s Axioms
1. To draw a straight line from any point to any point
2. To produce a straight line continuously in a straight line
3. To describe a circle with any center and radius
4. That all right angles are equal to one another
5. That, if a straight line falling on two straight lines makes the interior angles on the same side
less than two right angles, the two straight lines, if produced indefinitely, meet on that side
on which are the angles less than two right angles
Birkhoff ’s Axioms
Undefined Terms: point, line, distance, and angle
1. The points A, B, . . . of any line can be put into a one-to-one correspondence with the real
numbers, denoted A 7→ xA , so that |xb − xa | = d(A, B) for all points A and B.
2. One and only one line, `, contains any two distinct points P and Q.
3. The half-lines (or rays) `, m, n, . . . through any point O can be put into one-to-one correspondence with the real numbers a (mod 2π) so that if A and B are points other than O of ` and
m, respectively, the difference (am − an ) (mod 2π) of the numbers associated with lines ` and
m is m∠AOB (i.e. the measure of ∠AOB).
4. If in two triangles 4ABC and 4A0 B 0 C 0 and for some constant k > 0, d(A0 , B 0 ) = kd(A, B),
d(A0 , C 0 ) = kd(A, C), and m∠B 0 A0 C 0 = m∠BAC, then also d(B 0 , C 0 ) = kd(B, C), m∠C 0 B 0 A0 =
m∠CBA, and m∠A0 C 0 B 0 = m∠ACB.