MODERN EUCLIDEAN GEOMETRY

1. AN INTRODUCTION MODERN EUCLIDEAN GEOMETRY

2. MODERN EUCLIDEAN GEOMETRY • HILBERT’S MODEL • BIRKHOFF’S MODEL • SMSG (THE SCHOOL MATHEMATICS STUDY GROUP) POSTULATES

3. HILBERT’S MODEL

4. AXIOMS OF CONNECTION • Through any two distinct points A, B, there is always a line m. • Through any two distinct points A, B, there is not more than one line m. • On every line, there exist at least two distinct points. There exist at least three points which are not on the same line. • Through any three points not on the same line, there is one and only one plane.

5. THEOREM Two distinct line can not intersect in more than one point.

6. AXIOMS OF ORDER

7. ILUSTRATION OF THE 4TH AXIOM OF AXIOMS OF ORDER A m m P C B

8. THEOREMS • Theorem 1. Every line contains an infinity of points • Theorem 2. If A and B are points then there always exists a third point C such that A-C-B.

9. PROOF OF THEOREM 1

10. PROOF OF THEOREM 2

11. CONTINUATION OF PROOF OF THEOREM 2 By the second theorem of Axioms of Order, aw have E D B A

12. CONTINUATION OF PROOF OF THEOREM 2 By the 1st and 2nd theorem of Axioms of connection, and the 2nd theorem of Axioms of order, we have E D B F A

13. CONTINUATION OF PROOF OF THEOREM 2 E D C B A F

14. AXIOMS OF CONGRUENCE

15. THE 1ST AXIOM

16. THE 1ST AXIOM B A’ a a’ A B’ AB is congruent to A’B’

17. THE 2ND AXIOM If two line segments are congruent to third line segment then they are congruent to each other.

18. THE 3RD AXIOM

19. THE 4TH AXIOM

20. THE 5TH AXIOM