Proof of the Pythagorean theorem

History of the Pythagorean theorem

The Pythagorean theorem is one of the greatest scientific discovery of the human, and it is also one of the basic elementary geometry theorems. There are also many other names to call this theorem, like Shang-Gao theorem, Bai-Niu theorem and so on. Someone maybe will ask that what is the Pythagorean theorem. According to Wikipedia, the Pythagorean theorem “is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.[1]” This theorem has a very long history. Almost all ancient civilizations (Greece, China, Egypt, Babylon, India, etc.) have studied this theorem. In the West, this theorem was called Pythagorean theorem. According to legend, Pythagoras, an ancient Greek mathematician and philosopher, was the first person to discover this theorem in 550 BC. Unfortunately Pythagoras’ method of proving this theorem had been lost and we could not see how he proved now. But another famous Greek mathematician, Euclid (330 BC – 275 BC), gave us a good proof in his book called Euclid’s Elements. But Pythagoras was not the first person who discovered this theorem around the world. Ancient China discovered this theorem much earlier than him. So there is another name for the Pythagorean theorem in China, the Gou-Gu theorem. Zhong Jing is the first book about mathematics in China. And in the beginning of this book, there was a conversation between Zhong Gong and Shang Gao. They were talking about the way to solve the triangle problem. From this conversation, we could know that they already found out the Pythagorean theorem around 1100 BC. They found this theorem 500 years earlier than Pythagorean.

Proof of the Pythagorean theorem

Usually in a right triangle, we need to find the length of the third side when we already know the length of other two sides. For such problems, we can directly use the formula to calculate. In many problems, we need this theorem to solve many complex questions. And then, I will introduce two basic method to prove the Pythagorean theorem.

1) Proof by Zhao Shuang

In China, Zhang Shuang was the first person who gave us the earliest proof of the Pythagorean theorem. Zhao Shuang created a picture of “Pythagorean Round Square”, and used method of symbolic-graphic combination gave us a detailed proof of the Pythagorean theorem.

Assume a, b are two Right-angle side (b > a) and c is Hypotenuse. Then each area of a right triangle is equal to ab/2.

ZhaoShuangRtTriangleGraph

[Fig.1] Proof by Zhao Shuang

∵ RtΔDAH ≌ RtΔABE,

∴ ∠HDA = ∠EAB.

∵ ∠HAD + ∠HAD = 90º,

∴ ∠EAB + ∠HAD = 90º,

∴ ABCD is a square with side c, and the area of ABCD is equal to c2.

∵ EF = FG =GH =HE = DG―DH , ∠HEF = 90º.

∴ EFGH is also a square, and the area of ABCD is equal to (b-a)2.

∴ 4 *(1/2)(DG*DH)+(DG-DH)2=AD2

∴ DH2+DG2=AD2

2) Proof by Euclid

Just like we said before, Euclid gave us a good proof in his Euclid’s Elements. He also used method of symbolic-graphic combination.

In the first, we draw three squares and the side of each square are a, b, c. And then, let points H、C、B in a straight line. Next we draw two lines between F、B and C、D and draw a line parallel to BD and CE from A. This line will perpendicularly intersect BC and DE at K and L.

[Fig.2] Proof by Euclid

[Fig.2] Proof by Euclid

∵ BF = BA，BC = BD,

∠FBC = ∠ABD,

∴ ΔFBC ≌ ΔABD

∵ The area of ΔFBC is equal to (1/2)*FG2 and the area of ΔABD is half of the area of BDLK.

∴ The area of BDLK is equal to FG2. And then we can find the area of KLCE is equal to AH2 with the same method.

∵ The area of BDEC = The area of BDLK + The area of KLCE.

∴ FG2+AH2=BD2

Conclusion

The Pythagorean theorem’s development has exerted a significant impact on mathematics. And this theorem gave us an idea to solve geometric problems with Algebraic thinking. It is also a great example about symbolic-graphic combination. This idea is very important for solving mathematical problems. By the Pythagorean theorem, we can derive a number of other true propositions and theorems, which will greatly facilitate our understanding of geometry problems, but it also has driven the development of mathematics.

Reference

[1] http://en.wikipedia.org/wiki/Pythagorean_theorem

[Fig.1] https://upload.wikimedia.org/wikipedia/commons/e/e0/ZhaoShuangRtTriangleGraph.PNG

[Fig.2]https://upload.wikimedia.org/wikipedia/commons/5/59/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem2.svg