○ The Method for Finding the Inscribed Circle within a Right Triangle can be Known

The sum of the base, hypotenuse, and height is the key,
Combine the two numbers The base and the height to establish the divisor.
Multiply the base and height together; doubling this is the dividend.
Divide the doubled dividend by the divisor to find the circle’s diameter,
Let the calculator examine this closely and have no doubt.

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Now, suppose the base 勾 (Gou): the shorter horizontal leg of a right triangle is 27 feet and the height 股 (Gu): the longer vertical leg of a right triangle is 36 feet. We ask: what is the side length of the inscribed square 中容方面 (Zhong rong fang mian)?

The answer says: The side of the inscribed square is 15 feet, plus a remainder original: "有畸" (you ji), meaning there is a fractional part remaining..

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The Inscribed Square of a Right Triangle

[Diagram of a right-angled triangle with a square drawn inside it, sharing the right angle.] Height length: 36 feet Base length: 27 feet

Square side: 15 feet
Square side: 15 feet

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The method says: Set the base and height to be multiplied together, which results in 972 feet; this is the dividend 實 (Shi): the total value to be divided. Separately, take the base of 27 feet and add it to the height of 36 feet, totaling 63; this is the divisor 法 (Fa): the number used to divide the dividend. Divide the dividend by this. The resulting side length of the inscribed square is 15 feet. There is a remainder of 27, which is equivalent to 4 fen, 2 li, and 8 hao These are decimal units: 1 fen = 0.1, 1 li = 0.01, 1 hao = 0.001. The decimal result of 27/63 is approximately 0.428..

Now, suppose the base is 27 feet, the height is 36 feet, and the hypotenuse 弦 (Xian): the diagonal side is 45 feet. We ask: what is the diameter of the inscribed circle 中容圓徑 (Zhong rong yuan jing)?