Circle Inscribed in a Right Triangle

Height 股 (Gu) length: 36 feet Base 勾 (Gou) width: 27 feet Hypotenuse 弦 (Xian) length: 45 feet Inscribed circle diameter: 18 feet

How much [is the diameter]?

Answer: The inscribed circle diameter is eighteen feet.

The method states: Set the base of 27 feet and the height of 36 feet. Multiply them together to get 972 feet. Double this to get 1,944 feet; this is the dividend 實 (Shi): the total value to be divided. Add the base, height, and hypotenuse together for a total of 108; this is the divisor 法 (Fa): the number used to divide the dividend. Divide the dividend [by the divisor] to get the inscribed circle diameter of eighteen feet. This matches the question.

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Also: A Square Inscribed in a Right Triangle

Suppose there is a triangular block of jade, with a length of 1 foot 2 inches and a width of 6 inches. One wishes to cut the corner to form a square to create a seal. What is the side length of the square? Height length: 1 foot 2 inches In traditional Chinese measurement, 1 foot equals 10 inches; thus, this is 12 inches. Base width: 6 inches

Square side: 4 inches; Base segment: 4 inches; Height segment: 4 inches

Answer: Four inches.

The method states: Multiply the base and height together to get 72 inches; this is the dividend. Add the base and height together to get 18; this is the divisor. Divide them, and you will obtain the result. The formula used here is (Base × Height) ÷ (Base + Height) = Side of the inscribed square.

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