Zhao Shuang's Diagram of Strings

A tool made for my middle school daughter to help her understand the Pythagorean theorem.

The Pythagorean theorem, also known as Pythagoras' theorem, has many proofs. A Chinese proof is known as Zhao Shuang's Diagram of Strings.

Zhao Shuang's Diagram of Strings is a geometric figure presented by the ancient Chinese mathematician Zhao Shuang (also known as Liu Hui) in his annotations of the Nine Chapters on the Mathematical Art. This diagram, through intuitive geometric construction, demonstrates the relationship between the three sides of a right-angled triangle.

Core Structure:

1. Base Figure: A right-angled triangle forms the base; its two legs (gou, gu) and hypotenuse (xian) respectively constitute the sides of squares.
2. Assembly Method: Four congruent right-angled triangles and a smaller square are joined to form a larger square. The Pythagorean theorem is derived through the relationship of their areas.

Proof Process:

• Area Method: The side length of the larger square is c, and its area is c². The area of each triangle is ½*ab; the side length of the smaller square is b-a, and its area is (b-a)².

  The sum of the areas of the four right-angled triangles, 4*(1/2)*ab, and the smaller square, (b-a)², equals the area of the larger square, c².

  That is, 4*(1/2)*ab + (b-a)² = c² (a, b are the legs, and c is the hypotenuse)

  Through algebraic manipulation, we obtain a²+b²=c².

Zhao Shuang's Diagram of Strings is not merely a mathematical tool; it also reflects ancient China's understanding of the combination of geometry and algebra and is still used in mathematics education today.