1. The Theorem Of Pythagoras Brief description and proof of the Pythagorean theorem by dissection, based on squares of sum and difference M6) The theorem of pythagoras. Pythagoras of Samos was a Greek philosopher who lived http://www-spof.gsfc.nasa.gov/stargaze/Spyth.htm

Site Map (M-6) The Theorem of Pythagoras Pythagoras of Samos was a Greek philosopher who lived around 530 BC, mostly in the Greek colony of Crotona in southern Italy. According to tradition he was the first to prove the assertion (theorem) which today bears his name: If a triangle has sides of length ( a,b,c ), with sides ( a,b ) enclosing an angle of 90 degrees ("right angle"), then a + b = c A right angle can be defined here as the angle formed when two straight lines cross each other in such a way that all 4 angles produced are equal. The theorem also works the other way around: if the lengths of the three sides ( a,b,c ) of a triangle satisfy the above relation, then the angle between sides a and b must be of 90 degrees. For instance, a triangle with sides a b c = 5 (inches, feet, meterswhatever) is right-angled, because a + b = 9 + 16 = 25 = c Ancient Egyptian builders may have known the (3,4,5) triangle and used it (with measured rods or strings) to construct right angles; even today builders may still nail together boards of those lengths to help align a corner. Many proofs exist and the easiest ones are probably the ones based on algebra, using the elementary identities discussed in the preceding section, namely

2. Pythagorean Problem A method of disproving the theorem of pythagoras is presented. The author is adamant that this is intended only as a puzzle to find the mistake in the arguments, and not as a serious proposal. http://www.geocities.com/ResearchTriangle/System/8956/problems/pyth.htm

PLEASE NOTE: The following work is presented as a mathematical puzzle. It is NOT a valid proof, but serves to illustrate the problems that can arise if one is not familiar with postulates and conditions of various theorems. Read it and try to find the problem, but PLEASE do not preach to the world that Pythagoras' Theorem is false. A Disproof of Pythagoras' Theorem The Theorem of Pythagoras In a right triangle, the sum of the squares of the lengths of the two side sides is equal to the square of the hypotenuse. a + b = c DISPROOF: Start by defining a coordinate system with a along the x-axis and b along the y-axis. Let y = f(x) define the hypotenuse. Furthermore define a sequence of functions f n n n (x) converges uniformly to f(x). Clearly the length of the path defined by f (x) is a+b (or length a depending upon exactly how defines the path). Similarly, for any value of n the length of the path defined by f n (x) is also a+b. Since the functions f n (x) converge uniformly to f(x) the length of the path defined by f(x) is a+b.

3. Theorem Of Pythagoras Pythagore Demonstration of the theorem. http://www.alphaquark.com/Traduction/Pythagore.htm

Pythagore Demonstration of the theorem homepage Source of this page Author : Thibaut Bernard Number of visitor Update: Sunday 30 May 2001. Alphaquark author's Note : This page is a translation of with the help of Altavista translation I hope this translation is good, but if there are any errors, you can write me If this translation is successful, perhaps I will try to translate another document of Alphaquark Construction of the geometrical figure which will be used for the demonstration Let us take a rectangle of width A and height B. This rectangle which we make swivel of 90 o in the following way : For each rectangle, let us divide into two in the following way : Let us make swivel of 90 o the right-angled triangles in the following way yellow and purple : We thus find ourselves with four right-angled triangles. We note that one finds oneself with a square inside another. Demonstration Notation Let us take again our last diagram to indicate each of with dimensions by the following letters: One has four right-angled triangles of which : the with dimensions one opposed by a

4. Project MATHEMATICS!--Theorem Of Pythagoras The theorem of pythagoras. Video Segments. 1. Three questions from real life.2. Discovering the theorem of pythagoras. 3. Geometric interpretation. http://www.projectmathematics.com/pythag.htm

The Theorem of Pythagoras Video Segments Three questions from real life Discovering the Theorem of Pythagoras Geometric interpretation Pythagoras Applying the Theorem of Pythagoras Pythagorean triples The Chinese proof Euclid's elements Euclid's proof A dissection proof Euclid's Book VI, Proposition 31 The Pythagorean Theorem in 3D Contents The program begins with three real-life situations that lead to the same mathematical problem: Find the length of one side of a right triangle if the lengths of the other two sides are known. The problem is solved by a simple computer-animated derivation of the Pythagorean theorem (based on similar triangles): In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. The algebraic formula a + b = c is interpreted geometrically in terms of areas of squares, and is then used to solve the three real-life problems posed earlier. Historical context is provided through stills showing Babylonian clay tablets and various editions of Euclid's Elements . Several different computer-animated proofs of the Pythagorean theorem are presented, and the theorem is extended to 3-space.

5. The Theorem Of Pythagoras The theorem of pythagoras. Several engaging animated proofs of the Pythagorean theorem are presented with applications to reallife problems and to Pythagorean triples. http://www.maa.org/pubs/books/tpyvid.html

The Theorem of Pythagoras Several engaging animated proofs of the Pythagorean theorem are presented with applications to real-life problems and to Pythagorean triples. The theorem is extended to 3-space, but does not hold for spherical triangles. (22 minutes) List: $34.90 for video and workbook Catalog Code for Video and Workbook: TPYVID/W Workbook sold separately: $4.95 Catalog Code for Workbook: TPYWO/W How to Order Go to Subject Index Go to Publications Search © 1998 The Mathematical Association of America Please send comments, suggestions, or corrections about this page to webmaster@maa.org.

6. The Theorem Of Pythagoras (M6) The theorem of pythagoras. Pythagoras of Samos was a Greek philosopher wholived around 530 BC, mostly in the Greek colony of Crotona in southern Italy. http://www-istp.gsfc.nasa.gov/stargaze/Spyth.htm

Site Map (M-6) The Theorem of Pythagoras Pythagoras of Samos was a Greek philosopher who lived around 530 BC, mostly in the Greek colony of Crotona in southern Italy. According to tradition he was the first to prove the assertion (theorem) which today bears his name: If a triangle has sides of length ( a,b,c ), with sides ( a,b ) enclosing an angle of 90 degrees ("right angle"), then a + b = c A right angle can be defined here as the angle formed when two straight lines cross each other in such a way that all 4 angles produced are equal. The theorem also works the other way around: if the lengths of the three sides ( a,b,c ) of a triangle satisfy the above relation, then the angle between sides a and b must be of 90 degrees. For instance, a triangle with sides a b c = 5 (inches, feet, meterswhatever) is right-angled, because a + b = 9 + 16 = 25 = c Ancient Egyptian builders may have known the (3,4,5) triangle and used it (with measured rods or strings) to construct right angles; even today builders may still nail together boards of those lengths to help align a corner. Many proofs exist and the easiest ones are probably the ones based on algebra, using the elementary identities discussed in the preceding section, namely

7. Theorem Of Pythagoras This java applet shows you (automatically step by step) How ancient Chinesepeople discovers the same theorem. (much earlier than Pythagoras). http://www.phy.ntnu.edu.tw/~hwang/abc/Pythagoras.html

Theorem of Pythagoras a + b = c This java applet shows you (automatically - step by step) How ancient Chinese people discovers the same theorem. (much earlier than Pythagoras). You can change the interval delta T (in second, default value = 2 second). Click mouse button for manual control mode : Click right mouse button : show the following step Click left mouse button : show the previous step When you reach the last step, Press reset button to restart related Pythagoras java applet Your suggestions are highly appreciated! Please click hwang@phy03.phy.ntnu.edu.tw Author¡G Fu-Kwun Hwang Dept. of physics National Taiwan Normal University Last modified :¡@

8. Pythagoras (ii) The theorem of pythagoras for a right angled triangle the square onthe hypotenuse is equal to the sum of the squares on the other two sides. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

Pythagoras of Samos Born: about 569 BC in Samos, Ionia Died: about 475 BC Click the picture above to see eleven larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure. What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance. Pythagoras's father was Mnesarchus ([12] and [13]), while his mother was Pythais [8] and she was a native of Samos. Mnesarchus was a merchant who came from Tyre, and there is a story ([12] and [13]) that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited Italy with his father.

9. PinkMonkey.com Geometry Study Guide - 6.2 The Theorem Of Pythagoras 6.2 The theorem of pythagoras. Figure 6.3. D ABC is a right triangle. Index. 6.1The Right Triangle 6.2 The theorem of pythagoras 6.3 Special Right Triangles. http://www.pinkmonkey.com/studyguides/subjects/geometry/chap6/g0606201.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Get a scholarship! Need help with your research paper? See What's New on the Message Boards today! Favorite Link of the Week. ... Request a New Title Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey 6.2 The Theorem of Pythagoras Figure 6.3 D ABC is a right triangle. l (AB) = c l (BC) = a l (CA) = b CD is perpendicular to AB such that D ABC ~ D CBD or l (BC) l (AB) l (CD) a = c x = cx D ABC ~ D ACD or l (AC) l (AB) l (AD) b = c Therefore, from (1) and (2) a + b = cx + cy = c ( x + y ) = c c = c a + b = c The square of the hypotenuse is equal to the sum of the squares of the legs. Converse of Pythagoras Theorem : In a triangle if the square of the longest side is equal to the sum of the squares of the remaining two sides then the longest side is the hypotenuse and the angle opposite to it, is a right angle. Figure 6.4

10. PinkMonkey.com Geometry Study Guide - CHAPTER 6 : THEOREM OF PYTHAGORAS AND THE CHAPTER 6 theorem of pythagoras AND THE RIGHT TRIANGLE. Index. 6.1 The RightTriangle 6.2 The theorem of pythagoras 6.3 Special Right Triangles. Chapter 7. http://www.pinkmonkey.com/studyguides/subjects/geometry/chap6/g0606101.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Get a scholarship! Need help with your research paper? See What's New on the Message Boards today! Favorite Link of the Week. ... Request a New Title Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey CHAPTER 6 : THEOREM OF PYTHAGORAS AND THE RIGHT TRIANGLE 6.1 The Right Triangle Figure 6.1 D ABC is a right triangle, hence m ABC = 90 . Therefore m A and m C are complementary ( figure 6.1). Now seg.BD is a perpendicular onto seg.AC (figure 6.2). Figure 6.2 Seg.BD divides D ABC into two right triangles D BDC and D ADB ( figure 6.2). It can be easily proven that these two triangles are similar to the parent D ABC and therefore similar to each other. Proof : Consider D ABC and D BDC ABC BCA D ABC ~ D BDC Similarly consider D ABC and D ADB.

11. Theorem Of Pythagoras The theorem of pythagoras Pythagoras (fl. 500 BCE). The theorem of pythagoras is one of the earliest andmost important results in the history of mathematics. theorem of pythagoras. http://webphysics.ph.msstate.edu/javamirror/ntnujava/abc/Pythagoras.html

Theorem of Pythagoras a + b = c This java applet shows you (automatically - step by step) How ancient Chinese people discovers the same theorem. (much earlier than Pythagoras). You can change the interval delta T (in second, default value = 2 second). Click mouse button for manual control mode : Click right mouse button : show the following step Click left mouse button : show the previous step When you reach the last step, Press reset button to restart related Pythagoras java applet Your suggestions are highly appreciated! Please click hwang@phy03.phy.ntnu.edu.tw Author¡G Fu-Kwun Hwang Dept. of physics National Taiwan Normal University Last modified :¡@

12. The Theorem Of Pythagoras ... Key To Proof Use Menus, The key to the proof of the theorem of pythagoras. Thetheorem states that the area of the large white square (square of http://www.math.uwaterloo.ca/navigation/ideas/grains/pythagoras-key.shtml

University of Waterloo math@waterloo.ca prospective students current students alumni ... Use Menus The key to the proof of the theorem of Pythagoras The theorem states that the area of the large white square (square of the hypoteneuse) in the following diagram is equal to the sum of the areas of the two squares (of the two sides) in the next diagram It's important to notice that the orange triangle in both pictures is the same one, just rotated to better show the squares (and to match the animation). The proof begins by changing the solid white square to a blue one outlined in white. Then the same orange triangle is placed at each side of the square. The inside edges of the four triangles form the hypoteneus square. The outside edges of these four triangles form a large outer square. The outer square has fixed area equal to the area of the blue hypoteneuse square plus the areas of all four orange triangles. As the orange triangles move about in the animation, the outer square is preserved and consequently, the area of the outer square does not change.

13. Pythagorean Theorem -- From MathWorld Dixon, R. The theorem of pythagoras. §4.1 in Mathographics. New York Dover,pp. 9295, 1991. Project Mathematics. The theorem of pythagoras. Videotape. http://mathworld.wolfram.com/PythagoreanTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... Triangle Properties Pythagorean Theorem For a right triangle with legs a and b and hypotenuse c Many different proofs exist for this most fundamental of all geometric theorems. The theorem can also be generalized from a plane triangle to a trirectangular tetrahedron , in which case it is known as de Gua's theorem . The various proofs of the Pythagorean theorem all seem to require application of some version or consequence of the parallel postulate : proofs by dissection rely on the complementarity of the acute angles of the right triangle, proofs by shearing rely on explicit constructions of parallelograms, proofs by similarity require the existence of non-congruent similar triangles, and so on (S. Brodie). Based on this observation, S. Brodie has shown that the parallel postulate is equivalent to the Pythagorean theorem.

14. Template theorem of pythagoras a 2 + b 2 = c 2. This step). How ancient Chinesepeople discovers the same theorem. (much earlier than Pythagoras). http://www.phys.hawaii.edu/~teb/java/ntnujava/abc/Pythagoras.html

Theorem of Pythagoras a + b = c This java applet shows you (automatically - step by step) How ancient Chinese people discovers the same theorem. (much earlier than Pythagoras). You can change the interval delta T (in second, default value = 2 second). Click mouse button for manual control mode : Click right mouse button : show the following step Click left mouse button : show the previous step When you reach the last step, Press reset button to restart related Pythagoras java applet Your suggestions are highly appreciated! Please click hwang@phy03.phy.ntnu.edu.tw Author¡G Fu-Kwun Hwang Dept. of physics National Taiwan Normal University Last modified :¡@

15. Keymath.com : Discovering Geometry : JavaSketchpad™ Activities : The Theorem Of Home Discovering Geometry Dynamic Geometry Explorations The theorem of pythagoras.Dynamic Geometry Exploration. The theorem of pythagoras. http://www.keymath.com/DG/dynamic/pythagorean_theorem.html

Home Discovering Geometry Dynamic Geometry Explorations : The Theorem of Pythagoras Dynamic Geometry Exploration The Theorem of Pythagoras The Pythagorean Theorem relates the lengths of the three sides of a right triangle. In the investigation on this web page, you will learn more about the Pythagorean Theorem and see whether or not it works for triangles that are not right triangles. This investigation will help you understand Lesson 9.1 on pages 462-464 of Discovering Geometry: An Investigative Approach. Sketch The sketch below shows a right triangle with squares constructed on its three sides. You can drag vertex A to change the size and shape, but it will remain a right triangle. Sorry, this page requires a Java-compatible web browser. Investigate Click "Construct Center of Square" to show center O of the square on the longer leg. The center of a square is located at the intersection of the diagonals. Click "Construct j and k" to show two lines through O : line j perpendicular to the hypotenuse and line k perpendicular to line j . Lines j and k divide the square on the longer leg into four quadrilaterals. Click "Construct Quadrilaterals" to show the quadrilaterals and the square on the shorter leg.

16. BrochureWeb 18 THE theorem of pythagoras, DIRECTOR, AWARDS, Tom M. Apostol, GoldApple, 1989 National Educational Film and Video Festival, Oakland http://www-sfb288.math.tu-berlin.de/VideoMath/VideoMathReel/page19.html

18 T HE T HEOREM OF P YTHAGORAS D IRECTOR A WARDS Tom M. Apostol Gold Apple, 1989 National Educational Film and Video Festival, Oakland; Gold Medal, 1988 International Film and TV Festival of New York C ONTRIBUTORS Computer Animation - James F. Blinn Associate Producer - Joe Corrigan Narrator - Al Hibbs F URTHER I NFORMATION www.projmath.caltech.edu P RODUCER C ONTACT Tom M. Apostol Tom M. Apostol Project MATHEMATICS! 305 South Hill Avenue Pasadena, CA 91106, USA Tel: +1.626.395.3759 Fax: +1.626.395.3763 apostol@caltech.edu S UMMARY Several engaging animated proofs of the Pythagorean theorem are presented, with applications to real-life problems and to Pythagorean triples. The theorem is extended to 3-space, but does not hold for spherical triangles.

17. The Theorem Of Pythagoras. The theorem of pythagoras. Theorem 14.2.1. Remark 14.2.2 The ndimensional theoremof Pythagoras If are such that (ie ), then. . Noah Dana-Picard 2001-02-26. http://ndp.jct.ac.il/tutorials/alg-tut-win/node55.htm

The theorem of Pythagoras. Theorem 14.2.1 Remark 14.2.2 The n -dimensional theorem of Pythagoras: If are such that (i.e. ), then Noah Dana-Picard

18. 1300 A.C.: Theorem Of Pythagoras 1300 bC. theorem of pythagoras. In the tablet, of which the translationis reported, it seems really that the theorem of pythagoras is applied. http://web.genie.it/utenti/i/inanna/livello2-i/susa-pitagora-i.htm

HISTORY PHILOSOPHY RELIGION SCIENCE ... VERSIONE ITALIANA 1300 b.C. THEOREM OF PYTHAGORAS In the tablet, of which the translation is reported, it seems really that the theorem of Pythagoras is applied. In fact the calculation of the sides of a rectangle is exactly performed beginning from the knowledge of the diagonal (0,6666) and of the relationship existing between the width (W) and the length (L): W=L-L/4. Place: Susa (Mesopotamia) Epoch: 1300 b.C. - End of the I Dynasty of Babylon Tablet of Susa Problem We set that: - the width (of the rectangle) measures a quarter less in relationship to the length. Width = Length - Length/4 - the dimension of the diagonal is 0,6666. Diagonal = 0,6666 Which are the length and the width of the rectangle? Solution Set 1, the length, set 1 the prolongation. Arbitrary length = 1 0,25, the quarter, subtract from 1, you find 0,75. Arbitrary width = 1 - 0,25 = 0,75 Set 1 as length, set 0,75 as width, square 1, the length, you find 1. Square 0,75, the width, you find 0,5625.

19. Spacelink - The Theorem Of Pythagoras 9-12 912 The theorem of pythagoras 9-12. Project Mathematics! The Theoremof Pythagoras . Target Grades 9-12. Length 2030. This video http://spacelink.nasa.gov/NASA.News/NASA.Television.Schedules/Education.Schedule

Where am I? NASA Spacelink Home The Library NASA News NASA Television Schedules ... NASA TV Educational Programs The Theorem of Pythagoras 9-12 The Theorem of Pythagoras 9-12 Project Mathematics! "The Theorem of Pythagoras" Target: Grades 9-12 Length: 20:30 This video explains the Pythagorean Theorem and shows real life problems that can be solved using the Pythagorean Theorem. The program also illustrates several different animated proofs and weaves a historical perspective. Options Top of Page Educational Services Instructional Materials ... NASA Spacelink Home NASA Spacelink is a service of the Education Division of the National Aeronautics and Space Administration.

20. E.W. Dijkstra Archive: On The Theorem Of Pythagoras (EWD 975) On the theorem of pythagoras. For the theorem of pythagoras, I startfrom Coxeter s formulation ( Introduction to Geometry , p.8). http://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html

On the theorem of Pythagoras For the theorem of Pythagoras, I start from Coxeter's formulation ("Introduction to Geometry", p.8) "In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the two catheti." A formal expression of Coxeter's formulation is + b = c Elementary arithmetic yields the equivalent formulation + b = c Isn't that nicely symmetirc? It immediatly suggests - at least to me - the strengthening + b = c (This will turn out to be a theorem.) We get an equivalent formulation by negating both sides: + b + b and + b Bold perhaps, but not unreasonable. Note that (0), (1) and (2) are not independent: from any two of them, the third can be derived. They can be jointly formulated in terms of the function sgn - read "signum" - given by + b - c Consider now the following figure. We have drawn the case α + β < γ, in which the triangles ΔCKB and ΔAHC, of disjoint areas, don't cover the whole of ΔACB; denoting the area of ΔXYZ by "XYZ" we have in this case CKB + AHC = ACB

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