Pythagorean theorem is also called quotient height theorem, Pythagorean theorem or Pythagorean theorem.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of two right angles. If the two right angles of a right triangle are A and B and the hypotenuse is C, then A? +b？ =c？ That is, α * α+b * b = c * C

Summary: When the exponent is changed to n, the equal sign becomes less than the sign.

1. Origin of Pythagorean Theorem

According to textual research, human beings have known this theorem for at least 4000 years!

The first chapter of China's earliest mathematical work "Weekly Parallel Calculations" contains the relevant contents of this theorem: Duke Zhou asked: "I heard that doctors are good at counting, so I want to ask the ancients to set up a calendar of weeks and days." The sky cannot rise step by step, and the earth cannot be measured. How many times can I go out? "Shang Gao replied:" The counting method comes from the circle, and the circle comes from the square, and the square comes from the moment, and the moment is 998 1. Therefore, the moment is considered as three, the stock is four and the diameter is five. Outside is the square, half an hour, a circle is * * *. If you get that the moments of three, four, five and two are twenty and twenty-five, this is called product moments. Therefore, Yu rules the world because this number is born. "That is to say, a rectangle folded diagonally is called a right triangle. If the hook (short right side) is 3 and the rope (long right side) is 4, then the chord (hypotenuse) must be 5. From the above conversation, we can clearly see that people in ancient China discovered and applied Pythagorean theorem, an important mathematical principle, thousands of years ago.

The earliest documents in the west proved to be given by Pythagoras. It is said that when he proved Pythagorean theorem, he was ecstatic and killed a hundred cows to celebrate. Therefore, western countries also call Pythagorean Theorem "Hundred Cows Theorem". Unfortunately, Pythagoras' proof method has long been lost, and we have no way of knowing his proof method.

In fact, in earlier human activities, people have realized some special cases of this theorem. In addition to the above two examples, it is said that the ancient Egyptians also used the law of "hooking three strands, four strings and five" to determine the right angle. However, this legend has aroused the suspicion of many mathematical historians. For example, Professor M. Klein, an American mathematical historian, once pointed out: "We don't know whether the Egyptians realized the Pythagorean theorem. We know that they have rope puller (surveyor), but they tied a knot on the rope, divided the whole length into 3, 4 and 5 sections, and then used them to form a right triangle, which has never been confirmed in any literature. " However, archaeologists discovered several pieces of ancient Babylonian clay tablets, which were completed around 2000 BC. According to expert research, one of them is engraved with the following question: "A stick with a length of 30 units stands upright on the wall. How far is its lower end from the corner when its upper end slides down by 6 units? " This is a special case of a triangle with a ratio of three sides of 3:4:5. Experts also found that there was a strange number table carved on another clay tablet, in which * * * was engraved with four columns and fifteen rows of numbers, which was a Pythagorean number table: the rightmost column was the serial number from 1 to 15, while the left three columns were the values of strands, hooks and strings respectively, and a * * recorded/kloc. This shows that Pythagorean theorem has actually entered the treasure house of human knowledge.

Pythagorean theorem is a pearl in geometry, which is full of charm. For thousands of years, people have been eager to prove it, including famous mathematicians, painters, amateur mathematicians, ordinary people, distinguished dignitaries and even the president of the country Perhaps it is precisely because Pythagorean theorem is important, simple, practical and more attractive that it has been repeatedly 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. (The detailed proof of Pythagorean theorem is not included because the proof process is complicated. ※.)

People are interested in Pythagorean theorem because it can be generalized.

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.

And so on.