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