MAT 360 Lecture 5

1. MAT 360 Lecture 5 • Hilbert’s axioms - Betweenness

2. EXERCISE: • Can you deduce from the Incidence Axioms that there exist one point and one line? • Can you deduce from the Euclid’s I to V Axioms that there exist one point and one line? 2

3. Incidence Axioms • For each point P and for each point Q not equal to P there exists a unique line incident with P and Q. • For every line T there exist at least two distinct points incident with T. • There exist three distinct points with the property that no line is incident with all the three of them.

4. Euclid’s postulates • For every point P and every point Q not equal to P there exists a unique line l that passes for P and Q. • For every segment AB and for every segment CD there exists a unique point E such that B is between A and E and the segment CD is congruent to the segment BE. • For every point O and every point A not equal to O there exists a circle with center O and radius OA • All right angles are congruent to each other • For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.

5. Hilbert’s Axioms Note: you need to read all Chapter 3 while we work on it. Every statement previously proved in the text can be used • Incidence • Betweenness • Congruence • Continuity • Parallelism

6. Notation • By • A*B*C • we will mean “the point B is between the point A and the point C.”

7. AXIOMS OF BETWEENNESS (first part) • B1: If A*B*C then A, B and C are three distinct points lying on the same line and C*B*A. • B2: Given two distinct points B and D, there exist points A, C and E lying on BD such that A*B*D, B*C*D and B*D*E. • B3: If A, B and C are distinct points lying on the same line, then one and only one of the points is between the other two.

8. EXERCISES • Write the axiom B3 using the notation * we’ve just introduced. • Can you find a model for the Betweeness Axioms? • Consider a sphere S in Euclidean three-space and the following interpretation: A point is a point on S, a line is a great circle on S and incidence is set membership. Is this intrepretation a model of Betweeness Axioms? (What about Incidence Axioms?)

9. Old definitions revisted • The segment AB is the set of all points C such that A*C*B together with the points A and B. • The ray AB is the set of points on the segment AB together with all the points C such that A*B*C.

10. EXERCISE • Let A and B denote two points. Prove that • AB ∩ BA = AB • AB U BA = AB

11. Definition • Let l be a line. Let A and B be points not lying on l. • We say that A and B are on the same side of l if A=B or the segment AB does not intersect l. • We say that A and B are on opposite sides of l if A ≠ B and the segment AB does intersect l.

12. Questions • Suppose you have two points A and B lying on a line l. • Are A and B on the same side of l or on opposite sides of l? • Suppose you have two points A lying on a line l and B not lying on l. • Are A and B on the same side of l or on opposite sides of l?

13. AXIOMS OF BETWEENNESS (second part) • B4: For every line l and for every three points A, B and C not lying on l, • If A and B are on the same side of l and B and C are on the same side of l, then A and C are on the same side of l. • If A and B are on opposite sides of l and B and C are on opposite sides of l then A and C are on the same side of l.

14. Proposition • If A and B are on opposite sides of l and B and C are on same side of l then A and C are on opposite sides of l.

15. Definition: • A set of points S is a half plane bounded by a line l if there exists a point A such that S consists in all the points B for which A and B are on the same side of l.

16. Propositions • Every line bounds exactly two half planes and these two have planes have no point in common. • If A*B*C and A*C*D then B*C*D and A*B*D. • If A*B*C and B*C*D then A*B*C and A*C*D • (line separation property) If C*A*B and l is the line through A, B and C then for every point P lying on l, P lies either on the ray AB or on the ray AC

17. Pasch Theorem • If A, B and C are distinct noncollinear points and l is any line intersecting the line AB in a point between A and B, then l intersects either AC or BC. If C does not lie on l then l does not intersect both AC and BC.

18. Proposition • If A*B*C then B is the only point lying on the rays BA and BC and AB=AC.

19. Definition • A point D is in the interior of an angle <CAB if • D is on the same side of the line AC as B and • D is on the same side of the line AB as C.

20. Definition • The interior of a triangle is the intersection of the interior of its three angles. • A point P is exteriorto a triangle if it is not an interior point of a triangle and does not lie in any side of the triangle.

21. Proposition • If D is in the interior of <CAB then • Every point in the ray AD except A is in the interior of <CAB • None of the points in the ray opposite to the ray AD are in the interior of <CAB • If C*A*E then B is in the interior of <DAE

22. Definition • Ray AD is between rays ACand AB if AB and AC are not opposite rays and D is interior to <CAB.

23. Crossbar theorem • If the ray AD is between rays AC and AB then AD intersects segment BC

24. EXERCISE (18, Chapter 3) Consider the following interpretation. • Points: points (x,y) in the Euclidean plane such that both coordinates, x and y, have the form a/2n • Lines: Lines passing through several of those points. • Show that • The incidence axioms hold • The first three betweenness axioms hold. • Line separation property holds. • Pasch theorem fail • What about Crossbar theorem?