How did The Nine Chapters on the Mathematical Art solve right-triangle problems?

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East Asian mathematics Counting boards and markers, or counting rods, were used in China to solve systems of linear equations. This is an example from the 1st century ce.

In The Nine Chapters on the Mathematical Art, right-triangle problems are addressed through the use of algorithms rather than formal proofs. The text presents the Pythagorean theorem in this algorithmic form to solve various geometric challenges involving right triangles.1

For example, the text provides specific procedures to determine unknown dimensions when given related values, such as finding the height and hypotenuse when the base and the sum of the height and hypotenuse are known. Additionally, the work includes algorithms for calculating the diameter of an inscribed circle and the side length of an inscribed square within a right triangle.1