Zhao Shuang was an ancient Chinese mathematician who lived in the 3rd century CE. He spent his life on the study of the book called “Ling Xian” from famous Chinese scientist Zhang Heng and the ancient arithmetic book called “Zhou bi”. Zhao Shuang’s main achievement was summarizing the ancient mathematical achievements about “Gou gu mathematics” from the Han dynasty. Meanwhile he rediscovered the “Gou gu therorem”, which is actually the Chinese version of the “Pythagorean theorem”. He wrote hundreds of articles about “Gou gu geometric figures”, which are the earliest records of using several propositions to prove the sum-difference relations between the three sides of a triangle. In addition, he explored the roots of the quadratic equation, and he even put forward some conclusions, which are similar to Vieta’s formula (see the last link).
In Zhao Shuang’s articles about “Gou gu geometric figures”, Zhao Shuang used a method called the “cutting and compensation principle” sophisticatedly. And the meaning of “cutting and compensation principle” is actually: from 2ab+(b − a)2=c2, we can get a2 +b2 =c2. Its basic idea is that after appropriate “cutting” and “compensating” the area of the figure will not change. He also gave several propositions about the relations of the three edges of a triangle, such as:
(2(c − a)(c − b))1/2+(c-b)=a, (2(c − a)(c − b))1/2+(c-a)=b, (2(c − a)(c − b))1/2+(c-a)+(c-b)=c. Here c is the length of hypotenuse.
The most famous achievement of Zhao Shuang is an article with five hundreds words, which gives a brief proof of the “Gou gu theorem”. In that, he used figures to prove this theorem. And here I give two ways to show that.
Proof 1: Set a right triangle. Its two legs have lengths a and b respectively (assume b is larger than a) and the hypotenuse has length c. (In ancient Chinese we call the leg with length a as “gou”, leg with length b as “gu”, and the hypotenuse as “xian”, that’s why we call this theorem as “Gou gu theorem.”) Then four of such right triangles combine with a small square whose edge length is (b-a) to make a big square whose edge length is c. The small square is the white inner part of the picture below. Then apparently the area of the big square is c2, which can also be seen as the sum of the areas of the triangles and the small square. So we can get that 4ba/2 + (b − a)2=c2, from this we can get that c2=a2+b2. Thus it shows 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.
Proof 2: When a square with the edge length c (see the picture in the below, the large white square is just this square) is surrounded by 4 right triangles (see the 4 black right triangles in the picture), which are the same as the right triangles in the last proof, they form a bigger square with the edge length b+a. Then the area of the bigger square can be seen in two ways, the first way is (b − a)2. The second way is the sum of the area of the small square and the 4 right triangles. Thus we can get that (b − a)2= c2 +4ba/2, then we can get that c2=a2+b2.
These two proofs use the figures to show the “Gou gu theorem”, and both of them are very brief. They reflect the wisdom of ancient Chinese people. Also they are one of the great achievements of Zhao Shuang.