Proof 4: Liu Hui's Dissection (Cut-and-Paste)

A dissection proof attributed to Liu Hui (3rd century AD). Starting with side-by-side squares $a^2$ and $b^2$, we cut two triangles and translate them to the top to form a single tilted square $c^2$. This visually demonstrates $a^2 + b^2 = c^2$.

Prerequisites: pythagorean-theorem · Difficulty: intermediate

This elegant "cut-and-paste" proof is attributed to the 3rd-century Chinese mathematician Liu Hui. It demonstrates the Pythagorean theorem (Gougu theorem) by physically rearranging the squares $a^2$ and $b^2$ into a single square $c^2$.

The Concept

1. Start with the squares $a^2$ and $b^2$ placed side-by-side on a common baseline.
2. Cut two specific right triangles from the arrangement:
   • A triangle (base $a$, height $b$) from the $b^2$ square.
   • A triangle (base $b$, height $a$) spanning across the $a^2$ square.
3. Translate these triangles to the top of the figure.

The rearranged pieces fit perfectly together to form a single, tilted square. Since area is preserved during the rearrangement, and the side length of the new square is the hypotenuse $c$, we have visually proved: $a^2 + b^2 = c^2$.

The Proof

Step 1: Initial Squares

Start with square $b^2$ (blue) and square $a^2$ (red) placed side-by-side on a common baseline. Total area = $a^2 + b^2$.

Step 2: Pythagorean Theorem

Proof strategy (Liu Hui's Dissection). Cut two specific right triangles from the side-by-side $a^2 + b^2$ arrangement and translate them upward. The pieces rejoin to form a tilted $c^2$ square; since cutting and translating preserves area, $a^2 + b^2 = c^2$.

Step 3: The Cuts

Cut two specific right triangles from the arrangement:

• Blue Triangle from the $b^2$ square (base $a$, height $b$)
• Red Triangle spanning the $a^2$ square (base $b$, height $a$)

Step 4: Transformation

Translate the triangles to the top of the figure:

• Blue Triangle moves to the top of the $a^2$ square
• Red Triangle moves to the top of the $b^2$ square

Step 5: Result: $c^2$ Square

The pieces rejoin to form a perfect tilted square. The side length of this square is the hypotenuse $c$. Since area is conserved: $a^2 + b^2 = c^2$.

Conclusion

The pieces rejoin to form a perfect square tilted at an angle. The side length of this square is the hypotenuse $c$. Since area is conserved when we cut and rearrange pieces:

a^2 + b^2 = c^2
    

Historical Note

This proof is attributed to Liu Hui (3rd century AD), who wrote a commentary on The Nine Chapters on the Mathematical Art. His text describes "vermilion" (red) and "indigo" (blue) squares being cut and moved according to the "In-Out Complement Principle" (出入相補) — that area is conserved when a figure is dissected and rearranged.

Similar dissection proofs were later independently developed by the Arab mathematician Thabit ibn Qurra (9th century), demonstrating the universality of this visual logic.