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 a² and b², we cut two triangles and translate them to the top to form a single tilted square c². This visually demonstrates a² + b² = c².

Prerequisites: pythagorean-theorem · Difficulty: intermediate

Liu Hui's "cut-and-paste" proof rearranges the squares $a^{2}$ and $b^{2}$ into a single square $c^{2}$ on the hypotenuse:

a^{2} + b^{2} = c^{2}

Strategy: place the squares $a^{2}$ (red) and $b^{2}$ (blue) side-by-side, cut two right triangles from the arrangement, and translate them upward. The pieces rejoin into a tilted square of side $c$ , and since cutting and translating preserves area, $a^{2} + b^{2} = c^{2}$ .

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: $a^{2} + b^{2} = c^{2}$ .

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. ∎