Among more than 500 proofs of Pythagorean theorem, one is the proof of Garfield, the 20th president of the United States, which is simple and convenient. His proof method is also known as the "presidential proof method" in the history of mathematics.

Garfield proved this way: put two right-angled triangles with the same size together as shown in figure 1, that is, you get a right-angled trapezoidal ABCD. What is the area of right-angled trapezoidal ABCD at this time? (a+b)2=？ (a2+2ab+b2), where the area of right-angled trapezoidal ABCD is S △ AED+S △ EBC+S △ CED =? ab+？ ba+？ c2=？ (2ab+c2), so there is one? 2+b2=c？ 2, so the Pythagorean theorem is proved.

Second, the square area proof method of Zhao Shuang, a famous mathematician in China.

Zhao Shuang, a famous mathematician in China in the third century, proved in his book Pythagoras Square that this is a relatively simple and beautiful proof.

As shown in Figure 2, there are four congruent right-angled triangles with hypotenuse C in a square with side length C, and their right-angled sides are called A and B. This figure is used to prove the Pythagorean theorem.

Proof? Because the side length of the big square is C, the area of the big square is c2.

A big square consists of four congruent right-angled triangles and a small square in the middle, so the area of the big square = 4×? Ab+(a-b) 2 = A2+B2。 So A2+B2 = C2.

Third, take down the matchbox and verify the Pythagorean theorem.

Operation: As shown in Figure 3, set up a rectangular matchbox ABCD on the desktop. If you gently push it to the right from the upper end of the AB side, the matchbox will fall to the right around point C (at this time, the matchbox will not slide), and a horizontally placed matchbox A'B'CD will be obtained.

Observation: △ABC is a right triangle, and ∠ B = 90. If AB = B, BC = A and AC = C, then A2+B2 = C2.

Authentication: connect AC, A'c and AA'. It is easy to get AC⊥a'c from the knowledge of triangle congruence.

Because of the trapezoid ABD'A? The area of =? (A′D′+AB)？ BD′=？ (a+b)(b+a)=？ (a+b)2。

But trapezoidal ABD'A? Area = sδ ABC+sδ A ′ CD ′+sδ ACA ′ =? ab+？ ba+？ c2。

Namely. (a+b)2=？ ab+？ ba+？ c2。

So A2+B2 = C2.

Fourth, Euclid's proof method

In the Elements of Geometry, Euclid gave an extremely ingenious proof of Pythagorean theorem. Because of its beautiful figure, some people call it "the friar's headscarf" and others call it "the bride's sedan chair", which is really interesting. Professor Hua once suggested sending this character into the universe to communicate with "aliens".

Let's take a look at Euclid's verification method:

As shown in Figure 4, make three squares with side lengths of A, B and C, and put them in the shape shown in the figure, so that H, C and B are in a straight line, connecting BF and CD. C is CL⊥DE, AB is at point M, and DE is at point L? Because of AF? = AC, AB = AD, ∠ Fab = ∠ Gad, so δFab can be regarded as the rotation of△ △CAD around point A, because the area of△ Fab is equal to? The area of a2, △GAD is equal to half the area of rectangular ADLM, so the area of rectangular ADLM = a2. Similarly, the area of the rectangle MLEB = b2. The area of square ADEB = the area of rectangular ADLM+the area of rectangular MLEB, so C2 = a2+b2, that is, a2+b2 = c? 2.

I hope I can help you.