Pythagorean theorem is a basic geometric theorem, one of the important mathematical theorems discovered and proved by human beings in the early days, one of the most important tools to solve geometric problems with algebraic ideas, and one of the ties between number and shape. The ancient Babylonians knew and applied Pythagorean theorem as early as around 3000 BC, and they also knew many Pythagorean sequences. The ancient Egyptians also used Pythagorean theorem when they built magnificent pyramids and measured land after the Nile flooded. In China, the formula and proof of Pythagorean Theorem are recorded in Zhou Kuai Shu Jing, which is said to have been discovered by Shang Dynasty Shang Gao, so it is also called Shang Gao Theorem. Jiang Mingzu in the Three Kingdoms period made detailed comments on Pythagorean theorem in Jiang Mingzu's calculation, and gave another proof. The sum of the squares of the sides of two right angles (namely "hook" and "strand") of a right triangle is equal to the square of the side length of the hypotenuse (namely "chord"). That is to say, if the two right angles of a right triangle are A and B and the hypotenuse is C, then A? +b？ =c？ . There are about 400 ways to prove Pythagorean theorem, which is one of the most proven theorems in mathematics. Zhao Shuang gave a "Zhao Shuangxian Diagram" in the annotation of Zhou Bi suan Jing, which proved the accuracy of Pythagorean theorem. Pythagorean array equation a? + b？ = c？ Positive integer group (a, b, c) of. (3,4,5) is the Pythagorean number.

"Gousan, Gousi and Xian Wu" is one of the most famous examples of Pythagorean theorem. When integers a, b and c satisfy a? +b？ =c？ In this case, (a, b, c) is called a pythagorean array. That is to say, if the two right angles of a right triangle are A and B and the hypotenuse is C, then A? +b？ =c？ . "The common Pythagorean number is (3,4,5) (5, 12,13) (6,8, 10).

Pythagorean theorem is a basic geometric theorem. The sum of squares of the sides of a right triangle (hook and chord) is equal to the square of the side of the hypotenuse (chord). That is to say, if the two right angles of a right triangle are A and B and the hypotenuse is C, then A? +b？ =c？ . There are about 400 ways to prove Pythagorean theorem, which is one of the most proven theorems in mathematics. What is the Pythagorean number? +b？ =c？ Positive integer group (a, b, c) of. (3,4,5) is the Pythagorean number.

The proof method of Pythagorean theorem;

Garfield method

Five years after Garfield proved this conclusion, he became the 20th president of the United States, so people also called it the "presidential certificate law".

In right-angled trapezoidal ABDE, ∠ AEC = ∠ CDB = 90, △ AEC △ CDB,? ,? ,

Schematic diagram of "Presidential Certificate Law"

Garfield's variant

This proof is a variant of Garfield's proof

If you cut a big square with a side length of c diagonally, you will return to Garfield proof. On the contrary, if the two trapeziums in the above picture are put together, it becomes this proof method.

The area of the big square is equal to the area of the middle square plus the area of four triangles, namely:

Application of Pythagorean Theorem:

Xiao Ming is very happy after learning Pythagorean Theorem. He excitedly went home and told his father: in △ABC, if ∠ C = 90, BC=a, AC=b, AB=c, as shown in the following figure, according to Pythagoras theorem, A2+B2 = C2. Dad smiled and said, Very well, you have mastered the same knowledge. Now I will test you. If yes, please explain the reasons; If not, please compare Pythagorean theorem, try to guess the relationship between a2+b2 and c2, and prove your conclusion. (The picture below is for future use)

Answer:? Solution: ① When the triangle is an acute triangle,

Prove that the vertical foot of AD⊥BC is d and the length of CD is x,

According to Pythagorean Theorem, b2-x2=AD2=c2-(a-x)2.

Finishing: a2+b2=c2+2ax

2ax > 0

∴a2+b2>c2

② When the triangle is an obtuse triangle.

It is proved that the perpendicular passing through point B is AC and the length of CD is Y.

In the right triangle ABD, AD2=c2-(a+y)2.

In a right triangle ADC, AD2=b2-y2,

∴b2-y2=c2-(a+y)2

Finishing: a2+b2=c2-2ay

∵2ay>0,∴a2+b2 C2。

② In the obtuse triangle, A2+B2 < C2.

Analysis:? According to the meaning of the question, acute triangle and obtuse triangle should be proved separately and their heights should be made. According to the fact that the height is the common right-angled side of two right-angled triangles, it should be proved by Pythagorean theorem.