Definition
In a right angle triangle on the plane, the square of the two straight edges is equal to the square of the longeving long. If the two right side lengths of the right angle triangle are
The ticking theorem is a special case in the cosine ministry.
Decentralized, Zhao Shuang, Zhao Shuang, Zhao Shuang, Zhao Shuang, Zhao Shuang describes this picture: "The shares each multiplied, and it is mysterious. Fold, Xi Xuan. The case of Xuan map can be multiplied by Zhu Suri. Double it is Zhu Si. With the difference between the shares, it will be in the middle. The addition is also mysterious. Really subtract, half a short. With the difference as the method, the opening is removed, the resolution is hooked. Anything is hookped. Anyone is the real, that is, it is mystery. Or torque, or Outside. Symocation, body and numbers. The moment of challenge is wide, and Xuan is in Xuan. And the stock is solid. The momentum is hooked in mystery, and the rest is Shares. Both the stocks are from the two sides, the angle of the opening is the mystery. Additional stock is Xuan. With the difference of the shares, it is also the same as the shares. It is true. The result is also the law. The resulting also mysterious. It is the same as the law. The moment of share is in the mysteriousness of the hook, and the hook is. , The tensile stock is in mysterious, open the rest. Sailing on both sides, the angle of the opening, that is, the hook is poor. Hook in Xuan. Take the mystery with the different stocks. In addition to shares, it has to be hooked in. Let alone and share the same. Maximum. The resulting also mysterious. Share minus and self-employed, the method is hook, two differences, the result Take the shares of the shares. In the case of hitting the mysterious, it is increasing. Two differences are Xuan. Double Xuan is similar to the shares, see the consumer, take the picture, Double Xuan is full And there are many people, that is, the shares are similar. According to the difference, the rest is reduced, the rest, the outside. Extra-faced, that is, the shares, the shares, and the Xuan Siki is reduced. The rest, there is a yellow party. Huang Fang's face, the hook is poor. Take the difference and half of it. Plug in and half of the stock. It is a wide range of shares. Let the 袤 者 者 为In fact. Finally, the rest, the resulting result is the result. It is divided into half of the shortage in different points. Decreasing to Xuan immediately seek. "
Description by modern mathematics language is Huang Shi The area is equal to the area of the large square to reduce the area of four Chinese.
The meeting of the 24th International Mathematician Conference (ICM) is the figure.
Garfield Code
Garfield has become the 20th President of the United States, so it is also known as "Presidential Document".
In linear trapezoidal ABDE, ∠AEC = ∠CDB = 90 °, △ AEC △ CDB,
∵
Garfield Fair Variation
This proof is a variation of Garfield's law.
If a small square of the large square edge is cut along the diagonal, it is returned to the Garfield Fair. Conversely, if two trapezoids in the above figure are spent together, it will change to this prove.
The area of the large square is equal to the area of the intermediate square plus four triangles, namely:
Qing Zhu accessing Figure
Qing Zhu accessing map, is the end of the Eastern Mathematician Liu Hui according to "" The geometric certificate of the use of Number relationships "to prove the geometric certificate of the pyclotomy, distinctive characteristics, easy to understand.
Liu Hui describes this picture, "Hook all from Zhu Fang, the stocks will take into the Qing square, order to make a complement, each from its class, due to the rest of the other, the power of the synthesis The opening is, 即, ie, is also the case, one anyging triangle, with the width of the red square, Zhu Fang, as the stock leader is a cyan square. Arrange Zhu Fang and Qingfang two squares to the bottom of the bottom, and then make the virtual virtual, the split line does not move, and the line is "from its class", in the form of the synthesis, the string is the top. String length.
European miles
The following proof of the proof of the proof theorem is given in the "Geometric Original" book. Set △ ABC is a vertical angle triangle, where a is right angle. Painting from a , the line is straight to the side, which is perpendicular to the side. Extending the line into two squares on the side, the area is equal to the remaining two squares, respectively.
In this certification, we need the following four auxiliary theorem:
If two triangles have two groups of corresponding edges and the corners of the two sets of edges, Two-triangular shape. (SAS)
The triangular area is half of the parallelogram area of the same bottom.
Any square area is equal to its bilateral length product.
Any area of any rectangle is equal to its bilateral length product (according to auxiliary theorem 3).
European Delivery law (2 photos)
proof of proof is: from a point to the side to the side, make It is perpendicular to the opposite side. Extending the line into two squares on the side, two squares above, through the same triangle, by the equally high triangle, converted into the following two equivalent rectangles.set △ ABC as a vertical angle triangle, its right angle is ∠CAB.
It is BC, AB, and CA, which is sequentially plotted into quadangular CBDE, Bagf, and ACIH.
draws the BD, CE of the CE A, which is vertical BC and DE to K, L, respectively.
Connect the CF, AD, and form ΔBCF, ΔBda, respectively.
∠cab and ∠BAG are right angles, therefore C, A and G lines, truthable B, A, and H copies.
∠CBD and ∠FBA are right angles, so ∠ABD = ∠FBC.
Because AB = FB, BD = BC, △ ABD≌ △ FBC.
Because A and K and L are on the same straight line, quadrilateral BDLK = 2 △ ABD.
Because C, A and G are on the same straight line, the square BAGF = 2 △ FBC is formed.
therefore therefore quadrilateral BDLK = Bagf = AB2.
Similarly verified, quadrilateral ckle = acih = AC2.
add these two results, AB2 + AC2 = BD × BK + KL × KC
Due to BD = KL, BD × BK + KL × KC = BD (BK + KC) = BD × BC
Since CBDE is a square, AB2 + AC2 = BC2, ie A2 + B2 = C2.
This prove is proposed in the section 1.47 of the "Geometry original" book in Europe.
Due to the proof of this theorem depends on parallel axiom, many people question the necessary conditions for this theorem, until the 19th century attempt to negate the fifth axiom The European geometry appears.
promotion
数 数
The number of 股 定 定 定 <
Number of any set of tick
Theorem is used
Known right triangle two sides to solve the third side, or known the triangular length of the triangle, demonstrate that the triangle is a right-angle triangle or To prove that both sides of the triangle are vertical. The length of the tail length is the most basic application of the proimetary of the closure.
Simple history
China
BC 11 centuries, mathematician business high (Xe Zhou's first year) put forward "hook three, shares four, strings five". A period of "Zhou Yugui" written in the previous century ago has recorded a dialogue with Shanggong and Zhou Gong. Commercial height said: "...... 故 矩, hook three, stock repairs four, Jing Wu." It means: When the two right angles of the right angle triangle are 3 (hook) and 4 (shares), the diameter (Strove) is 5. In the future, people simply said this fact as "hook three-strand four string five". According to this exist, the 定 定 商 商 高 高 定 定 定 定 定 高 定 商 定 高 定
AD, Zhao Shuang in the Three Kingdoms era made detailed comments on the 勾 定 勾 定 勾 髀 于 于 于 于 于 于 于 于 于 于 于 于 于 于Square, ith chord, Zhao Shuang created a "clogging round square map", with a number combination to obtain a method, giving a detailed proof of the proof theorem. After Liu Hui, Liu Hui also proved the pyracketing theorem.
In the last year of China, Mathematics Home Huafang has put forward more than 20 kinds of 定 定 理 理.
Foreign
The Cuban Babini in about 3,000 BC will know and apply the ticking theorem, they also know many of the groups. The Cuban Babarba Plate called "Princeton 322" has been collected in the University of Columbia, which records a lot of tickings. The ancient Egyptians also used the Tsical Theorem for the Magnificent Pyramid and the land of the Nile River.
BA, Greek mathematician Pythagorass proves that the pyotonic theorem, and the Westerners are used to call this theorem for the Pydalas theorem.
The 4th century BC has given a certificate in "Geometry Original" (Volume I, proposition 47) in Greek.
On 1 April 1876, Gaird delivered his certificate for the proactive of the Characcery in "New England Education Log".
published 367 different licenses in 1940, "Picardo Las proposition".
meaning
1. The proof of the proof theorem is the origin of the geometry.
2. The ticking theorem is the first one in history that links the number and shape, that is, it is the first to link the geometry and the algebra.
3. The progenicle theorem has caused unreasonable discovery, causing the first mathematical crisis, greatly deepening the understanding of people.
4. The ticking theorem is the indefinite equation that has given a full solution in history, which leads to the Ma Ma's agency.
5. The ticking theorem is the basic theorem of the European geometry and has a huge practical value. This theorem not only is a glorious pearl in the geometry, is known as "the cornerstone of geometry", and has a wide range of applications in the field of higher mathematics and other science. On May 15, 1971, Nicaragua released a set of stamps entitled "Ten Mathematical Formulas that change the world", which is selected by the famous mathematician, and the chamfered theorem is one of them.