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Interactive Content

Liu Hui (225–295 CE), one of the great mathematicians of ancient China, urged his readers to make diagrams on paper and to cut and rearrange the pieces in order to justify mathematical statements. Today, an even simpler method of interactive mathematical demonstration can be used: the computer.

Explore the proof of the Pythagorean theorem via the xian tu diagram using the following sliders. This interactive exploration shows that while the xian tu is commonly shown for the special case of a (3, 4, 5)-triangle, the construction holds for arbitrary right triangles via the algebraic identity

4

1

2

ab+

2

(b-a)



2

a

+

2

b

, which in turn equals

2

c

:

Out[]=

​
a b

Now consider the special case of the (3, 4, 5)-triangle with

a3

,

b4

,

c5

and area

2

c

25

. A different dissection proof of the Pythagorean theorem follows by using the sliders to control rotations of two of the (3, 4, 5)-triangles. When fully rotated into place, the two rotated triangles plus two original triangles plus the central yellow square form a 3×3 square and a 4×4 square, thus demonstrating that

2

3

+

2

4



2

5

:

Out[]=

​
left triangle rotation right triangle rotation show squares

Computational Explanation

Recall the Pythagorean theorem, which states that the area of the square constructed on the hypotenuse of a right triangle is equal to the sum of the areas of squares constructed on its other two sides:

Out[]=

Consider a (3, 4, 5)-triangle—that is, a right triangle with legs of length

a3

and

b4

and hypotenuse of length

c5

:

Out[]=

By the Pythagorean theorem, the sum of the areas of squares erected on the legs is

2

3

+

2

4

9+1625,

which is equal to the area of a square erected on the hypotenuse, namely

2

5

25

.

The figure used to prove the theorem in Zhoubi Suanjing is usually referred to as the "hypotenuse diagram," or xian tu. It is ascribed to Zhen Luan (535–566), a Chinese mathematician and astronomer who mentions the diagram while discussing Zhao Shuang's third-century commentary of Zhoubi Suanjing. It provides an elegant dissection proof of the Pythagorean theorem, known in Chinese as the gou-gu theorem.

While the xian tu is commonly shown using the special case of a (3, 4, 5)-triangle:

Out[]=

… (left figure; from the Zhoubi Suanjing) the construction also works for an arbitrary right triangle (right figure). In particular, in the right diagram, note that the outer boundary consists of a square with side length c. Dissect this square into four congruent right triangles (shown in light-red) with side lengths a and b and hypotenuse c plus a middle square (shown in yellow) with side length

b-a

:

Out[]=

Adding together the areas of the four triangles plus that of the central square gives a total area of:

In[]:=

4

a×b

2

+

2

(b-a)

//Expand

Out[]=

2

a

+

2

b

Since the light-red triangles and yellow square exactly cover the outer square, their total area must equal the area of that square, namely

2

c

. This gives the identity

2

a

+

2

b



2

c

, proving the Pythagorean theorem.

Other Resources

Additional Reading

• Beckmann, P. A History of Pi. New York: Barnes & Noble, pp. 174–177, 1993.
• Cullen, C. "Zhao Shuang and Pythagoras's Theorem." Appendix One in Astronomy and Mathematics in Ancient China: The Zhou bi suan jing. Cambridge University Press, pp. 206–217, 1995.
• Dauben, J. W. "Chinese Mathematics." Ch. 3 in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 193–194, 214, 216–226 and 283, 2007.
• Gillon, B. S. "Introduction, Translation and Discussion of Chao Chun-ch'ing's [=Zhao Junqing = Zhao Shuang] 'Notes to the Diagrams of Short Legs and Long Legs and of Squares and Circles." Historia Mathematica, Vol. 4, pp. 253–293, 1977.
• Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. Princeton, NJ: Princeton University Press, pp. 248–261, 2011.
• Liu, H. Jiu zhang suan shu. 1896. [Photoreprint ed. of the Wenyuan Ge copy. Taipei: Taiwan shang wu yin shu guan, 1975.]
• Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 62–66, 2007.
• Martzloff, J.-C. A History of Chinese Mathematics. Berlin: Springer-Verlag, pp. 296–299, 1987.
• Shen, K.; Crossley, J. N.; and Lun, A. W.-C. The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford and Beijing: Oxford University Press, 1999.
• Swetz, F. and Kao, T. I. Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. University Park, PA: Pennsylvania University Press, 1977.

Additional Links

• Mathematical Treasures - Zhoubi suanjing
• Hathi Trust Digital Library: Jiu zhang suan shu xi cao tu shuo (1896)
• Hathi Trust Digital Library: Jiu zhang suan shu. v.1-3 (1899)
• Wikipedia: Pythagorean theorem
• Wikipedia: Zhoubi Suanjing