Download MAT360 Lecture 1 Euclid`s geometry

INTRODUCTION TO
Euclid’s geometry
The origins of geometry
A “jump” in the way of thinking geometry
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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)
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The axiomatic method
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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?
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What are the advantages of the empirical
method?
Undefined terms
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point,
line,
lie on,
between,
congruent.
More about the undefined terms
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By line we will mean straight line (when we
talk in “everyday” language”)
How can straight be defined?
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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
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For every point P and every point Q not
equal to P there exists a unique line l that
passes through P and Q.
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Notation: This line will be denoted by
More undefined terms
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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.

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Notation: This segment will be denoted by AB
Second Euclid’s postulate
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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.
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Another formulation
Let it be granted that a segment may be
produced to any length in a straight line.
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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
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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
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All right angles are congruent to each other.
Definition of parallel lines
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Two lines are parallel if they do not intersect,
i.e., if no point lies in both of them.
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If l and m are parallel lines we write l || m
Euclidean Parallel Postulate
(equivalent formulation)
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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.
II.
III.
IV.
V.
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.
Euclid’s postulates (another formulation)
Let the following be postulated:
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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
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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.
EXERCISE
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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
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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
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Things which equal the same thing also
equal to each other.
Exercise (Euclid’s proposition 1)
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Given a segment AB. Construct an
equilateral triangle with side AB.
Exercise. Prove the following using the
postulates
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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?
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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.