INTRODUCTION TOEuclid’s geometry

The origins of geometry

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)

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.

Question

What are the advantages of the axiomatic method?

What are the advantages of the empirical method?

Undefined terms

point, line, lie on, between, congruent.

More about the undefined terms By line we will mean straight line (when we

talk in “everyday” language”)

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)

“Carpenter’s meaning of straight”

Euclid’s first postulate

For every point P and every point Q not equal to P there exists a unique line l that passes through P and Q.

Notation: This line will be denoted by

More undefined terms

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

We will also use some “underfined terms” from set theory (for example, “intersect”, “included”, etc) All these terms can be defined with the above terms (set, being member of a set).

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

Second Euclid’s 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.

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.

Euclid’s postulate III

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

Definition

Definition of angle

Notation

We use the notation

for the angle with vertex A defined previously.

Questions

Can we use segments instead of rays in the definition of angles?

Is the zero angle (as you know it) included in the previous definition?

Are there any other angles you can think of that are not included in the above definition?

Definition

Definition of right angle.

Euclid’s Postulate IV

All right angles are congruent to each other.

Definition of parallel lines

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

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.

Euclid’s postulates (modern formulation)I. For every point P and every point Q not equal to

P there exists a unique line l that passes for P and Q.

II. 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.

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

IV. All right angles are congruent to each otherV. 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.

Euclid’s 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.

Exercise: Define

Midpoint M of a segment AB Triangle ABC, formed by tree noncollinear

points A, B, C Vertices of a triangle ABC. Define a side opposite to a vertex of a

triangle ABC.

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EXERCISE

Warning about defining the altitude of a triangle.

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

perpendicular bisector of AB.

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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.

Common notion

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

Exercise (Euclid’s proposition 1) Given a segment AB. Construct an

equilateral triangle with side AB.

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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.

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Second Euclid’s 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.