Pythagorean theorem is a pearl in geometry, so it is full of charm. For thousands of years, people have been eager to prove it, including famous mathematicians, amateur mathematicians, ordinary people and distinguished dignitaries. Perhaps it is precisely because of the importance, simplicity and attractiveness of Pythagorean theorem that it has been repeatedly hyped and demonstrated for hundreds of times. 1940 published a proof album of Pythagorean theorem, which collected 367 different proof methods. In fact, that's not all. Some data show that there are more than 500 ways to prove Pythagorean theorem, and only the mathematician Hua in the late Qing Dynasty provided more than 20 wonderful ways to prove it. This is unmatched by any theorem.

Method 1. Draw two squares with side length (a+b), as shown in the figure, where A and B are right-angled sides and C is hypotenuse. The two squares are congruent, so the areas are equal.

The left picture and the right picture each have four triangles that are the same as the original right triangle, and the sum of the areas of the left and right triangles must be equal. If all four triangles in the left and right images are deleted, the areas of the rest of the image will be equal. There are two squares left in the picture on the left, with A and B as sides respectively. On the right is a square with C as its side. therefore

a2+b2=c2 .

This is the method introduced in the geometry textbook. Intuitive and simple, everyone can understand. Method 2: Draw squares directly on three sides of a right triangle, as shown in the figure.

This proof method is wonderful because they use few theorems and only use two basic concepts of area:

(1) The area of congruence is equal;

⑵ Divide a graph into several parts, and the sum of the areas of each part is equal to the area of the original graph.

This is a completely acceptable simple concept that anyone can understand.

2 "the ancients" method:

As shown in the figure, the four right-angled triangles in the figure are painted deep red, the small square in the middle is painted white, and the square with the chord as the side is called chord solid. Then, after patchwork and matching, he affirmed that the relationship between the three pythagorean chords conforms to the pythagorean theorem. That is, "Pythagoras shares multiply each other, and they are real strings, and they are divided, that is, strings." Zhao Shuang's proof of Pythagorean theorem shows that China mathematicians have superb ideas of proving problems, which are concise and intuitive.

Euclid gave a generalization theorem of Pythagorean theorem in Elements of Geometry: "A straight side on the hypotenuse of a right triangle has an area equal to the sum of the areas of two similar straight sides on two right angles".

From the above theorem, the following theorem can be deduced: "If a circle is made with three sides of a right-angled triangle as its diameter, the area of the circle with the hypotenuse as its diameter is equal to the sum of the areas of two circles with two right-angled sides as its diameter".

Pythagorean theorem can also be extended to space: if three sides of a right triangle are used as corresponding sides to make a similar polyhedron, then the surface area of a polyhedron on the hypotenuse is equal to the sum of the surface areas of two polyhedrons on the right side.

If three sides of a right-angled triangle are used as balls, the surface area of the ball on the hypotenuse is equal to the sum of the surface areas of two balls made on two right-angled sides.

In a word, on the road of Pythagorean theorem exploration, we are moving towards the palace of mathematics.