Chapter 2: Euclid’s Proof of the Pythagorean Theorem

MATH 402ELAINE ROBANCHO

GRANT WELLER

Outline

Euclid and his ElementsPreliminaries: Definitions, Postulates, and

Common NotionsEarly PropositionsParallelism and Related TopicsEuclid’s Proof of the Pythagorean TheoremOther 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 BCE13 books on mathematics and geometryAxiomatic: began with 23 definitions, 5

postulates, and 5 common notionsBuilt these into 465 propositionsOnly the Bible has been more scrutinized

over timeNearly all propositions have stood the test of

time

Preliminaries: Definitions

Basic foundations of Euclidean geometryEuclid defines points, lines, straight lines,

circles, perpendicularity, and parallelismLanguage is often not acceptable for modern

definitionsAvoided using algebra; used only geometryEuclid never uses degree measure for angles

Preliminaries: Postulates

Self-evident truths of Euclid’s system

Euclid only needed fiveThings that can be done

with a straightedge and compass

Postulate 5 caused some controversy

Preliminaries: Common Notions

Not specific to geometrySelf-evident truthsCommon 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

SASASAAASSSSThese 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 triangleWant 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!

MathematicianMathematician ProofProof

Chou-pei Suan-ching (China), 3rd c. BCE

Bhaskara (India), 12th c. BCE

James Garfield (U.S. president), 1881

Other proofs of the Theorem

Further issues

Controversy over parallel postulateNobody could successfully prove itNon-Euclidean geometry: Bolyai, Gauss, and

LobachevskiGeometry where the sum of angles of a

triangle is less than 180 degreesGives you the AAA congruence