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# Proof without Words Pythagorean Theorem

### Proof without Words: The Pythagorean Theorem

The Pythagorean theorem is proved geometrically in yet another way. This article originally appeared as Proof without Words: a 2 + b 2 = c 2. A pdf copy of the article can be viewed by clicking below. Since the copy is a faithful reproduction of the actual journal pages, the article may not begin at the top of the first page Basically, it is a proof without words of the Pythagorean Theorem. First, you start with a right triangle. Then, you have the areas of each side: a^2, b^2, and c^2. To make things easy, I will call them Square A, Square B, and Square C Pythagorean Theorem Proofs Without Words. Perigal's Proof. Garfield's Proof. Puzzle/Dissection Proof. Pythagorean Theorem Proofs Without Words. Author: Vivian Buchanan. Explore 3 different picture proofs of the Pythagorean Theorem. Table of Contents. Perigal's Proof. Proof of Pythagoras' theorem (Henry Perigal

Pythagorean Theorem Proof Without Words (request for words) Ask Question Asked 9 years, 5 months ago. Active 7 years, 3 months ago. Viewed 7k times 32 5 $\begingroup$ I was intrigued by a book I saw called Proofs without Words. So I bought it, and discovered that the entire book doesn't have any words in it Proof Without Words: Pythagorean Theorem. Content Provider. Illuminations . Illuminations is a project designed by the National Council of Teachers of Mathematics (NCTM) and supported by the Verizon Foundation. Illuminations works to serve you by increasing access to quality standards-based resources for teaching and learning mathematics. Proof #36. Hi! This is Pythagorean proof #36 and it is quite simple! As you can see, the triangle in the middle is a right triangle, and the squares are the side lengths squared (hee hee!). This is really easy to understand, and here is how it works. The Pythagorean Theorem is (a x a) + (b x b) = (c x c) (sorry I can't do squared on this. Pythagorean Theorem Proof Without Words (request for words) 8. Naive approach to Pythagoras. 4. Pythagorean Theorem intuition. Related. 32. Pythagorean Theorem Proof Without Words (request for words) 1. Strange proof of Schwarz Inequality with Pythagorean Theorem. 2. Calculate the depth of water in the trough when it is exactly half full. 3

### Pythagorean Theorem Proof Without Words Mrs

• Warning! Pictures can be misleading! Theorem? Hmmm, it looks like 32:5 = 31:5. D.C. Ernst Proofs Without Words 2/1
• Unlike a proof without words, a droodle may suggest a statement, not just a proof. The Pythagorean configuration is known under many names, the Bride's Chair being probably the most popular. Besides the statement of the Pythagorean theorem, Bride's chair has many interesting properties, many quite elementary
• I threw together this video showing a visual proof without words of the Pythagorean Theorem.The background music is a cover of the overworld theme from Sup..

In mathematics, a proof without words is a proof of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous due to their self-evident nature Proof Without Words. Author: Steve Phelps. Topic: Geometry, Pythagoras or Pythagorean Theorem. Drag the points in the sketch below On Proofs Without Words Robin L. Miller Whitman College May 14th, 2012 Behold! 1 Introduction Most mathematicians will be familiar with the above picture. This diagram, credited to the Ancient Chinese mathematical text Zhou Bi Suan Jing, is a charmingly simple visual proof of the Pythagorean Theorem, one of mathematics' most fundamental.

Proof Without Words: The Pythagorean Theorem. John Molokach. Johnston Community College, Smithfield, NC 27577 Correspondence johnmolokach@gmail.com. View further author information. Page 287 | Received 02 Oct 2016, Accepted 12 Feb 2019, Published online: 19 Sep 2019. Page 287 There are also some excellent on-line interactive visual proofs of the Pythagorean Theorem that are of a distinctly more recent vintage. Consider, for example, in Figure 17 the PWW 2.0 rendering of Rufus Isaacs' Proof Without Words of the Pythagorean Theorem that we showed in Figure 1 of the Introduction. Figure 17 Pythagorean Theorem Algebra Proof What is the Pythagorean Theorem? You can learn all about the Pythagorean Theorem, but here is a quick summary:. The Pythagorean Theorem says that, in a right triangle, the square of a (which is a×a, and is written a 2) plus the square of b (b 2) is equal to the square of c (c 2): a 2 + b 2 = c 2. Proof of the Pythagorean Theorem using Algebr

(2020). Yet Another Proof Without Words of the Pythagorean Theorem. Mathematics Magazine: Vol. 93, No. 4, pp. 306-306 Dec 16, 2014 - This is a dynamic visual proof showing congruence through rigid transformations for the pythagorean theorem

Details. This proof without words of the Pythagorean theorem extends from Mamikon's theorem, which is a generalization of the strategy demonstrated here of equating a set of moving tangents (tangent sweep) to their parallel translations with a common end point (tangent cluster). In general, Mamikon's theorem says, The area of a tangent sweep of a space curve is equal to the area of its. This proof-without-words of the Pythagorean Theorem is far from a new one, but it's the first one I've ever seen 'in the wild' (this photo was snapped after finishing dinner at a Mongolian Grill restaurant) The Pythagorean theorem can be proven without words as shown in the second diagram on left. The two different methods for determining the area of the large square give the relation. between the sides. This proof is more subtle than the above, but still can be considered a proof without words

### Pythagorean Theorem Proofs Without Words - GeoGebr

1. ing the area of the large square give the relation + = between the sides. This proof is more subtle than the above, but still can be considered a proof without words
2. what we're going to do with this video is study a proof of the Pythagorean theorem that was first discovered first discovered as far as we know first discovered by James Garfield in 1876 and what's exciting about this is he was not a professional mathematician you might know James Garfield as the 20th president of the United States he was elected president he was elected four years after 18 in.
3. Proof Without Words: Pythagorean Theorem 6-8 This applet proves the Pythagorean Theorem without words using geometry
4. Dec 24, 2015 - Proofs Without Words - GeoGebra Book of Pythagorean Theorem proofs by Steve Phelps. Dec 24, 2015 - Proofs Without Words - GeoGebra Book of Pythagorean Theorem proofs by Steve Phelps. Pinterest. Today. Explore. When the auto-complete results are available, use the up and down arrows to review and Enter to select. Touch device.

Proof without words: | In |mathematics|, a |proof without words| is a |proof| of an identity or mathematical sta... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled Proof Without Words: The Pythagorean Theorem with Equilateral Triangles Claudi Alsina (claudio.alsina@upc.edu), Universitat Politecnica de Catalunya,` 08028 Barcelona, Spain, and Roger B. Nelsen (nelsen@lclark.edu), Lewis & Clark College, Portland, OR 97219 Ta T Tb Tc The Pythagorean theorem (Proposition I.47 in Euclid's Elements) i (2017). Proof Without Words: The Pythagorean Theorem. The College Mathematics Journal: Vol. 48, No. 5, pp. 334-334 Pythagorean Theorem Proof Without Words. New Resources. Types of Triangle & Sum of angle of Triangle; Absolute Value and Distanc

Pythagoras Theorem is an important topic in Maths, which explains the relation between the sides of a right-angled triangle. It is also sometimes called the Pythagorean Theorem. The formula and proof of this theorem are explained here with examples. Pythagoras theorem is basically used to find the length of an unknown side and angle of a triangle Given its long history, there are numerous proofs (more than 350) of the Pythagorean theorem, perhaps more than any other theorem of mathematics. The proofs below are by no means exhaustive, and have been grouped primarily by the approaches used in the proofs The two key facts that are needed for Garfield's proof are: 1. The sum of the angles of any triangle is 180 . 2. The area of a trapezoid with bases of length b1 and b2 and height h is A 1 2 b1 b2 h. Before giving Garfield's Proof of the Pythagorean Theorem, we will first give proofs of the above two facts. Pythagorean Theorem Proof Without Words. This puzzle is a great little project or activity to help students understand the Pythagorean Theorem! The proof could easily be added to an interactive notebook for foldable for students as well. I love proofs like this for geometry Proof without Words: Pythagorean Theorem at the National Council of Teachers of Mathematics Illuminations Web site provides a step-by step JAVA applet demonstrating the proof that is attributed to Bhaskara (see fig.4). The site also has detailed instructions and an exploration of the proof

Here is one of the shortest proofs of the Pythagorean Theorem. Suppose we are given any right triangle with sides of lengths A, B, C. In order to show that. A 2 + B 2 = C 2. it is enough to show for any set of three similar figures whose widths relate to each other in the proportions A:B:C, that the area of the largest figure is the sum of the. Wise,David S.: Proof without words: Generalization from Pythagoras,Mathematics Magazine,vol. 71,no.1(Feb 1998),p.64. On Internet . Pythagoras' Theorem, by Bill Casselman, The University of British Columbia. Many more interesting proofs by Alexander Bogomolny. A proof of the Pythagorean Theorem by Liu Hui (third century AD There are lots of geometrical proofs for e.g. the pythagorean theorem that need no words. It's not like code golf any more than a text-based proof is. Words still need to get interpreted just as images do. If anything, a succinct geometric proof can be much easier to grasp and verify than a text-based proof the Pythagorean Theorem. Dozens of very different proofs of the theorem have been devised, and many of them can be given in the form of proofs without words. The diagram to the right shows one of the most straightforward and accessible proofs (without words) of the Pythagorean Theorem 2+ = 2 PROOFS WITHOUT WORDS Proofs without words cover a wide range of mathe-matical concepts—including algebra, trigonometry, geometry, and calculus—and can be used in such courses as the history of mathematics. I generally introduce students to proofs without words by using those available on the Illuminations Web site

### geometry - Pythagorean Theorem Proof Without Words

Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. Pythagorean Theorem, Rational Irrational Numbers, Geometry With Practice, Tests, Answer Key For Homeschool or Classroom (160 pgs) 1,932. Paperback Proofs Without Words. Author: Steve Phelps. Topic: Geometry, Pythagoras or Pythagorean Theorem. Here is a GeoGebraBook of Proofs Without Words for the Pythagorean Theorem The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. This means that if the triangle has side lengths a, b and hypotenuse c, the following equation holds: c 2 = a 2 + b 2. Right Triangle ABC. The proof of the theorem is shown below D.C. Ernst Proofs without Words 11 / 17; Theorem (Pythagorean Theorem) If a, b, c ∈ N are the lengths of the sides of a right triangle, where c the length of the hypotenuse, then a2 + b2 = c2. D.C. Ernst Proofs without Words 11 / 17 D.C. Ernst Proofs without Words 12 / 17; Theorem We have the following fact concerning integrals: π/2 Proofs without words (PWWs) are figures or diagrams that help the reader see why a particular mathematical statement is true, and how one might begin to formally prove it true. PWWs are not new, many date back to classical Greece, ancient China, and medieval Europe and the Middle East

### Proof Without Words: Pythagorean Theorem - SA

• The Pythagorean Theorem I and (2) 17.Heo (2015) Proof Without Word: The Pythagorean Theorem. Make sense of one of the proofs provided in the readings. Submit your own drawing and explanation of the proof in your own words
• An arbelos is the figure bounded by these three semicircles. Draw the perpendicular to PR at Q, meeting the largest semicircle at S. Then the area A of the arbelos equals the area C of the circle with diameter QS [Archimedes, Liber Assumptorum, Proposition 4]. S S A C P Q R A=C Q Proof. A A1 B1 A2 A2 A1 B2 A + A1 + A2 = B1 + B2 B1 C1 C2 B2 B1.
• i-mysteries to be solved. The book starts with six different figures proving the Pythagorean Theorem -- and only one has any kind of language on it (a few variables and their products). These examples' author range from a 10th century Arab mathematician, a 3rd centur
• The Pythagorean Theorem (first of many proofs): the left diagram shows that , and the right diagram shows a second proof by re-arranging the first diagram (the area of the shaded part is equal to , but it is also the re-arranged version of the oblique square, which has area ). Another proof of the Pythagorean Theorem (animated version)
• And if you swap them out for copies of the original right triangle, you get a beautiful proof without words of the Pythagorean Theorem: There are hundreds of proofs of the Pythagorean Theorem (if a triangle is right, then a ² + b ² = c ²), but the pancake demo ask us to believe the converse (if a ² + b ² = c ², then the triangle is right)
• A collection of quality worksheets with variable problems for grades 3-8. Topics include angle relationships, triangles, quadrilaterals, congruency, similar figures, constructions, area, volume, and the Pythagorean Theorem. Price: \$9.00 download. See the free samples

### Proof #36: A Proof Without Words - Pythagorean Theorem

• Proof Without Words: Pythagorean Theorem. Click Image to Enlarge : Watch a dynamic, geometric proof without words of the Pythagorean Theorem. Can you explain the proof? SEE MORE : 6. Pythagorean Explorer. Click Image to Enlarge : Find the length of a side of a right triangle using the Pythagorean Theorem, and then check your answers
• ee, but did not meet the good article criteria at the time. There are suggestions below for improving the article. Once these issues have been addressed, the article can be reno
• Mathematical proof is strict and rigorous, however by using diagrams and graphs, it can also be beautiful. Some mathematical identities and relationships can be proven without words, which are simple and logical. Like Martin Gardner wrote, There is no more effective aid in understanding certain algebraic identities than a good diagram
• • Geometric Square Proof • Proof without Words • More Proofs • Euclid's Elements • Triples • Coordinate Geometry • Time Line. Brief History • Evidence suggests that the Pythagorean Theorem was known by all mathematical cultures well before the time of Pythagoras himself. Map of Locations Greece Egypt Mesopotamia India China
• PdF:MA2MP_SMR2 Metody řešení matematických úloh 2. Operace. Vyberte řádek zatržením vlevo. Vyhledat. Zobrazit ikony. Vyhledat. Omezit soubory a složky: jen složky/jen soubory. složky, které mají/nemají anglický název
• The Pythagorean Theorem rephrased using area! Now, if you look at our Pythagorean equation a2 + b2 = c2, you see three numbers being squared. In order to get to our picture proof, we need to interpret everything through pictures. Well, squared numbers make us think of squares, and area! In particular, a2 is the area of a square of.

### geometry - Pythagorean Theorem Proof Without Words 6

• ing the area of the large square give the relation + = between the sides. This proof is more subtle than the above, but still can be considered a proof without words
• Here are my top 7 ideas for the Pythagorean Theorem (in no particular order): 1 - Pythagorean Theorems Word Problems Coloring Worksheet - I love using this worksheet because it gets students to practice word problems, with less complaining because of the coloring. 2 - Pythagorean Theorem Proof Without Words - This post has a free download of a template for showing your students a visual.
• Media in category Proof without words The following 43 files are in this category, out of 43 total. AM GM inequality visual proof.svg 512 × 341; 2 KB. AM-GM inequality.svg 162 × 162; 211 bytes. Chineese Pythagorean theorem 2.PNG 928 × 531; 334 KB. Chinese pythagoras.jpg 871 × 475; 69 KB. Pythagorean proof.png 274 × 165; 8 KB.

### Pythagorean Theorem: A Proof Without Words Nicollier

(Making note of this algebraic identity without knowing the Pythagorean theorem might be a bit of a trick, but it doesn't seem unreasonable to think it possible.) (Note that as well, and this corresponds to the equation , where and are the legs of a right triangle with hypotenuse of length The command \newtheorem{theorem}{Theorem} has two parameters, the first one is the name of the environment that is defined, the second one is the word that will be printed, in boldface font, at the beginning of the environment. Once this new environment is defined it can be used normally within the document, delimited it with the marks \begin{theorem} and \end{theorem} 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Example 1: The Irrational Club wants to build a tree house. They have a 9-foot ladder that must be propped diagonally against the tree # mathematics # proof # pythagorean theorem # visual proof # pythagorean theorem # hello # hi # hey # welcome # flirting # photography # jobs # photojojo # hanson # #geometria #color #barbie_elektrix # homer simpson # season 5 # episode 10 # 5x10 # homer simpson # episode 15 # season 13 # 13x1

- theorem proof without words - this post has a free download of a template for showing your students a visual. This geometry worksheet theorem worksheet is suitable for - grade. in this theorem worksheet, students use the theorem to find the length of the unlabeled side of the right triangle Proofs Without Words III - March 2016. We use cookies to distinguish you from other users and to provide you with a better experience on our websites The design, called the husan-thu, is an ancient Chinese proof without words of the Pythagorean Theorem. In January 2010, the quilt was featured on the cover of The College Mathematics Journal . [back to quilting page

Pythagorean Theorem, true for any triangles. Ninth Century CE: Tabit ibn Qorra ibn Mervan, Abu-Hasan, al-Harrani, an Arabic mathematician, proves a generalization of the Pythagorean Theorem involving any triangle. Eleventh Century CE: Bhaskara's Behold! Proof. Bhaskara gives a proof without words of the Pythagorean Theorem adapted from Proof without Words: Exercises in Visual Thinking, by Roger B. Nelson, as shown in Winicki-Landman (1998, p. 723). 1/x 1/x x 1/x 1/x x x x Prove: If x > 1, then x + 1/x ≥2. We can construct a right triangle with the given sides so that it satisﬁes the Pythagorean theorem. x - 1 2 + 22 = x + 1 xx This statement is true, that. Proof without words: (Pythagorean theorem) โดย Frank Burk (1996) ในรูปสามเหลี่ยมมุมฉากที่มี a และ b เป็นความยาวของด้านประกอบมุมฉาก และ c เป็นความยาว.

Figure 3. A visual proof of the Pythagorean theorem. It is probably one of the rst proofs. Problem B: Use Figure (3) for a proof of the Pythagorean theorem. You can either describe in words, or label some parts of the picture. Remember that we want to show c 2= a + b2. ab a b a+b 2 Figure 4. A visual proof of p ab (a+ b)=2. 2 6.4 Figure 4.3 The proof of Pythagoras' Theorem by Euclid. Here is one proof of Pythagoras' Theorem in Euclid's Elements7. Consider any right triangle. Let us denote by c the length of its hypotenuse, and denote by a;b the lengths of other two sides, respectively. In Figure 4.3, in each of the two big squares, we see four equal size triangles So when you have 2 halves of 2 rectangles that make up a square, you get a half of a square. Then, since we started with 1/2 of the area of the triangles, 1/2a^2 + 1/2b^2 = 1/2c^2. Divide by 1/2 and you have the Pythagorean Theorem, a^2 = b^2 = c^2. QE 1. Draw four congruent right triangles. Congruent triangles are ones that have three identical sides. Designate the legs of length a and b and hypotenuse of length c. The Pythagorean Theorem states that the sum of squares of the two legs of a right triangle is equal to the square of the hypotenuse, so we need to prove a2 + b2 = c2

### Pythagorean Theorem and its many proof

2.1 Proof using similar triangles 2.2 Euclid's proof 2.3 Proof by rearrangement 2.4 Algebraic proofs 2.5 Proof using differentials 3 Converse 4 Consequences and uses of the theorem 4.1 Pythagorean triples 4.2 Incommensurable lengths 4.3 Complex numbers 4.4 Euclidean distance in various coordinate systems 4.5 Pythagorean trigonometric identit Our first post, the Pythagorean Theorem falls under Geometry and Trigonometry categories. Tags are the keywords that are related to our article. The possible tags for our post The Pythagorean Theorem are Pythagorean theorem, proof without words, Pythagoras, and right triangles. Notice that tags are more content-specific than categories. » Read. The theorem has been given numerous proofs - possibly the most for any mathematical theorem. With this puzzle you will be able to observe the geometric proof of the Pythagorean theorem. A great way to teach kids through practice and play. Triangle size: 9x12x15cm. Not only for kids, suitable for all ages James' answer is good and so is this one for the 3, 4, 5 case anyway. _____ Also this one is very impressive!!! A REAL MUST SEE!!! Pythagorean theorem water demo https://youtu.be/CAk.. 89.79 Proof without words: Pappus' generalisation of Pythagoras' theorem - Volume 89 Issue 516 Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites

### Pythagorean Theorem - Proof without words - YouTub

There are many proofs of the the Pythagorean Theorem. For example, an idea of proof is given by considering the pictures below (Rufus Isaac, Two Mathematical Papers without Words, Mathematics Magazine, Vol. 48 (1975), p. 198).Let us consider two congruent squares The Pythagorean theorem is one of the most fundamental result of all euclidean geometry 1 : Pythagoras' theorem: If is the right- angle triangle with legs , and hypotenuse then . In words: In a right-angled triangle the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. 2 I have seen a lot of teachers teach the Pythagorean Theorem in hundreds of different ways in my over a decade of teaching math. There is one however, that I will forever use! This lesson is on the proof of the pythagorean theorem. This is THEE GREATEST WAY to Teach the Pythagorean Theorem! (There is no close 2nd Place even! This is that good!

### Pythagorean Theorem - Proof Without Words - YouTub

Although Pythagoras' name is attached to this theorem, it was actually known centuries before his time by the Babylonians. There are many proofs of this theorem, some graphical in nature and others using algebra. See A graphical proof of the Pythagorean Theorem for one such proof.. On the web site cut-the-knot, the author collects proofs of the Pythagorean Theorem, and as of this writing has. Pythagorean Theorem Proof Thousands of proofs of this theorem exist, including one by U.S. president James Garfield (before he became president). One proof is easy to make with graph paper, a straightedge, pencil, and scissors Theorem (Law of Cosines). c^2 = a^2 + b^2 - 2ab Cos [C]. Proof. Place triangle ABC on a Cartesian coordinate system such that angle C is at the origin and length a lies on the x-axis. Then length b is on the other ray from the origin. We can easily identify the coordinates of two of the vertices: Vertex C lies at (0,0), and vertex B lies at (a,0)

The inaugural issue had a lot of material on the Pythagorean theorem and on themes related to this theorem. There was a 'proof without words' from the late Prof A R Rao; articles on paper folding, on the use of spreadsheets, on a way of classifying quadrilaterals, on the use of math portfolios in teaching, and on teaching fractions at the. So adding the areas of the four triangles and the inner square you get 4*1/2*a*b+ (b-a) (b-a) = 2ab +b^2 -2ab +a^2=a^2+b^2 which is c^2. It is much shorter that way. I figured it out in the 10th grade after seeing the diagram and knowing it had something to do with proving the Pythagorean Theorem 2. Proof without words: The difference of consecutive integer cubes is congruent to 1 modulo 6 (with C. Alsina and H. Unal), The College Mathematics Journal, 45 (2014), p. 135 PDF. 3. Proof without words: Pythagorean quadruples, The College Mathematics Journal, 45 (2014), p. 179 PDF 4 Pythagorean Theorem calculator to find out the unknown length of a right triangle. It can deal with square root values and provides the calculation steps, area, perimeter, height, and angles of the triangle. Also explore many more calculators covering math and other topics

Proof #36 This proof is due to J. E. Böttcher and has been quoted by Nelsen (Proofs Without Words II, p. 6). I think cracking this proof without words is a good exercise for middle or high school. While Proof #64 at cuttheknot advertises that Socrates' result -- the isosceles Pythagorean theorem -- leads to the full Pythagorean theorem, I would disagree with this analysis. The dissection given there is essentially a standard dissection that leads to the full Pythagorean theorem, with or without Socrates' result Which statement could you use to make the proof of Pythagorean Theorem using the diagram? answer choices . Because 3+4=7, the Pythagorean Theorem is a 2 +b 2 =c 2. Because 9+16=25, the Pythagorean Theorem is a+b=c. Because 3+4=7, the Pythagorean Theorem is a+b=c. Because 9+16=25, the Pythagorean Theorem is a 2 +b 2 =c 2 Pythagorean Quadratic MAT 221: Introduction to Algebra Pythagorean Quadratic The Pythagorean Theorem was termed after Pythagoras, who was a well-known Greek philosopher and mathematician, and the Pythagorean Theorem is one of the first theorems identified in ancient civilizations. The Pythagorean theorem says that in any right triangle the sum of the squares of the lengths of the legs is. Cut the Knot: Pythagorean Theorem and its Many Proofs: Written proofs. GeoGebra Book of Proofs Without Words for the Pythagorean Theorem by Steve Phelps is a series of manipulatives for exploration. National Library of Virtual Manipulatives: investigate methods for determining congruent triangles (Java required). Explore SSS, SAS, ASA, SSA 分类Proof without words中的媒体文件 AM GM inequality visual proof.svg 512 × 341；2 KB. AM-GM inequality.svg 162 × 162；211字节. Chineese Pythagorean theorem 2.PNG 928 × 531；334 KB.