After Liu Hui's commentary on the Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), c. 263 CE
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The square on the hypotenuse can be assembled, by sliding alone, from four congruent pieces of the smaller leg-square and one whole copy of the larger leg-square.
- On the two legs of the right triangle, build squares outward — on the longer leg, on the shorter.
- Mark the center of , the smaller square. Draw two lines through that point — one parallel to the hypotenuse, one perpendicular to it. Together they cut into four congruent pieces.
- is left whole. It becomes the fifth piece of the puzzle, untouched.
- This is the in-and-out principle (chū rù xiāng bǔ): whatever sticks out of one region is carried, by sliding alone, into a gap in another, until the target is exactly filled with nothing left over and nothing doubled up. Here, the five pieces slide — no turning — into place to exactly cover the square on the hypotenuse.
- Every scrap of area from the two leg-squares lands inside the hypotenuse's square, and none of it overlaps. So .
Therefore c² = a² + b². ∎
Liu Hui’s own diagram for this argument does not survive; his surviving commentary describes the principle of cutting and sliding pieces to match areas, without the figure itself. What’s shown here is a modern rendering built to satisfy that same rule honestly — every piece here moves by sliding only, verified by direct computation rather than by eye — not a claim to reproduce his lost original stroke for stroke. The four-piece cut works for any right triangle; quartering the smaller leg-square rather than the larger one, as done here, is just as valid a choice as the reverse.