Pythagorean Theorem: Euclid`s proof
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Transcript Pythagorean Theorem: Euclid`s proof
Chapter 2: Euclid’s Proof of the
Pythagorean Theorem
MATH 402
ELAINE ROBANCHO
GRANT WELLER
Outline
Euclid and his Elements
Preliminaries: Definitions, Postulates, and Common
Notions
Early Propositions
Parallelism and Related Topics
Euclid’s Proof of the Pythagorean Theorem
Other Proofs
Euclid
Greek mathematician – “Father
of Geometry”
Developed mathematical proof
techniques that we know today
Influenced by Plato’s
enthusiasm for mathematics
On Plato’s Academy entryway:
“Let no man ignorant of
geometry enter here.”
Almost all Greek
mathematicians following
Euclid had some connection
with his school in Alexandria
Euclid’s Elements
Written in Alexandria around 300 BCE
13 books on mathematics and geometry
Axiomatic: began with 23 definitions, 5 postulates,
and 5 common notions
Built these into 465 propositions
Only the Bible has been more scrutinized over time
Nearly all propositions have stood the test of time
Preliminaries: Definitions
Basic foundations of Euclidean geometry
Euclid defines points, lines, straight lines, circles,
perpendicularity, and parallelism
Language is often not acceptable for modern
definitions
Avoided using algebra; used only geometry
Euclid never uses degree measure for angles
Preliminaries: Postulates
Self-evident truths of
Euclid’s system
Euclid only needed five
Things that can be done
with a straightedge and
compass
Postulate 5 caused some
controversy
Preliminaries: Common Notions
Not specific to geometry
Self-evident truths
Common Notion 4: “Things which coincide with one
another are equal to one another”
To accept Euclid’s Propositions, you must be
satisfied with the preliminaries
Early Propositions
Angles produced by
triangles
Proposition I.20: any two
sides of a triangle are
together greater than the
remaining one
This shows there were
some omissions in his
work
However, none of his
propositions are false
Construction of triangles
(e.g. I.1)
Early Propositions: Congruence
SAS
ASA
AAS
SSS
These hold without reference to the angles of a
triangle summing to two right angles (180˚)
Do not use the parallel postulate
Parallelism and related topics
Parallel lines produce
equal alternate angles
(I.29)
Angles of a triangle sum
to two right angles (I.32)
Area of a triangle is half
the area of a
parallelogram with same
base and height (I.41)
How to construct a
square on a line segment
(I.46)
Pythagorean Theorem: Euclid’s proof
Consider a right triangle
Want to show a2 + b2 = c2
Pythagorean Theorem: Euclid’s proof
Euclid’s idea was to use areas of squares in the proof.
First he constructed squares with the sides of the
triangle as bases.
Pythagorean Theorem: Euclid’s proof
Euclid wanted to show that the areas of the smaller
squares equaled the area of the larger square.
Pythagorean Theorem: Euclid’s proof
By I.41, a triangle with the same base and height as
one of the smaller squares will have half the area of
the square. We want to show that the two triangles
together are half the area of the large square.
Pythagorean Theorem: Euclid’s proof
When we shear the triangle like this, the area does
not change because it has the same base and height.
Euclid also made certain to prove that the line along
which the triangle is sheared was straight; this was
the only time Euclid actually made use of the fact
that the triangle is right.
Pythagorean Theorem: Euclid’s proof
Now we can rotate the triangle without changing it.
These two triangles are congruent by I.4 (SAS).
Pythagorean Theorem: Euclid’s proof
We can draw a perpendicular (from A to L on
handout) by I.31
Now the side of the large square is the base of the
triangle, and the distance between the base and the
red line is the height (because the two are parallel).
Pythagorean Theorem: Euclid’s proof
Just like before, we can do another shear without
changing the area of the triangle.
This area is half the area of the rectangle formed by
the side of the square and the red line (AL on
handout)
Pythagorean Theorem: Euclid’s proof
Repeat these steps for the triangle that is half the
area of the other small square.
Then the areas of the two triangles together are half
the area of the large square, so the areas of the two
smaller squares add up to the area of the large
square.
Therefore a2 + b2 = c2 !!!!
Pythagorean Theorem: Euclid’s proof
Euclid also proved the converse of the Pythagorean
Theorem; that is if two of the sides squared equaled
the remaining side squared, the triangle was right.
Interestingly, he used the theorem itself to prove its
converse!
Other proofs of the Theorem
Mathematician
Chou-pei Suan-ching
(China), 3rd c. BCE
Bhaskara (India),
12th c. BCE
James Garfield (U.S.
president), 1881
Proof
Further issues
Controversy over parallel postulate
Nobody could successfully prove it
Non-Euclidean geometry: Bolyai, Gauss, and
Lobachevski
Geometry where the sum of angles of a triangle is
less than 180 degrees
Gives you the AAA congruence