Chapter 2: Euclids Proof of the Pythagorean Theorem

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Chapter 2 Euclids Proof of the Pythagorean
Theorem

• Math 402Elaine RobanchoGrant Weller

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Outline

• Euclid and his Elements
• Preliminaries Definitions, Postulates, and
  Common Notions
• Early Propositions
• Parallelism and Related Topics
• Euclids Proof of the Pythagorean Theorem
• Other Proofs

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Euclid

• Greek mathematician Father of Geometry
• Developed mathematical proof techniques that we
  know today
• Influenced by Platos enthusiasm for mathematics
• On Platos Academy entryway Let no man ignorant
  of geometry enter here.
• Almost all Greek mathematicians following Euclid
  had some connection with his school in Alexandria

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Euclids 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

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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

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Preliminaries Postulates

• Self-evident truths of Euclids system
• Euclid only needed five
• Things that can be done with a straightedge and
  compass
• Postulate 5 caused some controversy

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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 Euclids Propositions, you must be
  satisfied with the preliminaries

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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)

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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

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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)

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Pythagorean Theorem Euclids proof

• Consider a right triangle
• Want to show a2 b2 c2

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Pythagorean Theorem Euclids proof

• Euclids idea was to use areas of squares in the
  proof. First he constructed squares with the
  sides of the triangle as bases.

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Pythagorean Theorem Euclids proof

• Euclid wanted to show that the areas of the
  smaller squares equaled the area of the larger
  square.

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Pythagorean Theorem Euclids 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.

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Pythagorean Theorem Euclids 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.

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Pythagorean Theorem Euclids proof

• Now we can rotate the triangle without changing
  it. These two triangles are congruent by I.4
  (SAS).

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Pythagorean Theorem Euclids 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).

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Pythagorean Theorem Euclids 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)

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Pythagorean Theorem Euclids 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 !!!!

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Pythagorean Theorem Euclids 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!

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Other proofs of the Theorem

• Mathematician

• Proof

• Chou-pei Suan-ching (China), 3rd c. BCE
• Bhaskara (India),
• 12th c. BCE
• James Garfield (U.S. president), 1881

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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