MAT360 Lecture 1 Euclids geometry

1
MAT360 Lecture 1Euclids geometry

• The origins of geometry

2
Homework due next Thursday

• Chapter 1 Exercises 1,2,3,4, Mayor Exercise 1.
• To think (but not submit) Exercises 8 and 11,
  and how can straight be defined.

3
A jump in the way of thinking geometry

• Before Greeks experimental

• After Greeks Statements should be established by
  deductive methods.
• Thales (600 BC)
• Pythagoras (500 BC)
• Hippocrates (400 BC)
• Plato (400 BC)
• Euclid (300 BC)

4
The axiomatic method

• A list of undefined terms.
• A list of accepted statements (called axioms or
  postulates)
• A list of rules which tell when one statement
  follows logically from other.
• Definition of new words and symbols in term of
  the already defined or accepted ones.

5
Question

• What are the advantages of the axiomatic method?
• What are the advantages of the empirical method?

6
Undefined terms

• point,
• line,
• lie on,
• between,
• congruent.

7
More about the undefined terms

• By line we will mean straight line (when we talk
  in everyday language)

8
How can straight be defined?

• Straight is that of which the middle is in front
  of both extremities. (Plato)
• A straight line is a line that lies symmetrically
  with the points on itself. (Euclid)
• Carpenters meaning of straight

9
Euclids first postulate

• For every point P and every point Q not equal to
  P there exists a unique line l that passes for P
  and Q.
• Notation This line will be denoted by

10
More undefined terms

• Set
• Belonging to a set, being a member of a set.

11
Definition

• Given two points A and B, then segment AB between
  A and B is the set whose members are the points A
  and B and all the points that lie on the line and
  are between A and B.
• Notation This segment will be denoted by AB

12
Second Euclids postulate

• 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.
• Another formulation
• Let it be granted that a segment may be produced
  to any length in a straight line..

13
Definition

• Give two points O and A, the set of all points P
  such that the segment OP is congruent to the
  segment OA is called a circle. The point O is the
  center of the circle. Each of the segments OP is
  called a radius of the circle.

14
Euclids postulate III

• For every point O and every point A not equal to
  O there exists a circle with center O and radius
  OA.

15
Definition
16
Question

• What terms are defined in the previous slide?

17
Definition
18
Definition
19
Notation
20
Question

• Is the zero angle included in the previous
  definition?

21
Definition
22
Definition
23
Euclids Postulate IV

• All right angles are congruent to each other.

24
Definition

• Two lines are parallel if they do not intersect,
  i.e., if no point lies in both of them.
• If l and m are parallel lines we write l m

25
Euclidean Parallel Postulate (equivalent
formulation)

• 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.

26
Euclids postulates (modern formulation)

• 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.

27
Euclids postulates (another formulation)Let the
following be postulated

• Postulate 1. To draw a straight line from any
  point to any point.
• Postulate 2. To produce a finite straight line
  continuously in a straight line.
• Postulate 3. To describe a circle with any center
  and radius.
• Postulate 4. That all right angles equal one
  another.
• Postulate 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 the
  two right angles.

28
Exercise Define the following terms

• Midpoint M of a segment AB
• Triangle ABC, formed by tree noncollinear points
  A, B, C.
• Warning about defining the altitude of a
  triangle.

29
Exercise. Prove the following using the postulates

• For every line l, there exists a point lying on l
• For every line l, there exists a point not lying
  on l.
• There exists at least a line.
• There exists at least a point.

30
Exercise

• Prove using the postulates that if P and Q are
  points in the circle OA, then the segment OP is
  congruent to the segment OQ.

31
Common notion

• Things which equal the same thing also equal to
  each other.

32
Exercise (Euclids proposition 1)

• Given a segment AB. Construct an equilateral
  triangle with side AB.

33
(No Transcript)
34
Second Euclids postulates Are they equivalent?

• 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.
• Any segment can be extended indefinitely in a
  line.

35
Exercise

• Define lines l and m are perpendicular.
• Given a segment AB. Construct the perpendicular
  bisector of AB.