Haidao Suanjing/Examples/Example 1
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Example of Problem from Haidao Suanjing by Liu Hui
- What is the size of a square inscribed in the corner of a right-angled triangle to touch the hypotenuse?
Solution
Let the lengths of the legs of the given right-angled triangle be $a$ and $b$.
Then the length of the side of the inscribed square is $\dfrac {a b} {a + b}$.
Proof
Let $x$ be the length of the side of the inscribed square be $x$.
Without loss of generality, let $a < b$.
Let the right-angled triangle be half of a rectangle whose sides are of length $a$ and $b$.
Let the rectangle be dissected along the straight lines shown.
Let the pieces of the dissection be assembled into a rectangle whose sides are of length $a + b$ and $x$.
Then we have:
- $a b = x \paren {a + b}$
and the result follows.
$\blacksquare$
Sources
- 263: Liu Hui: Haidao Suanjing
- 1987: Li Yan and Du Shiran: Chinese Mathematics: A Concise History (translated by John N. Crossley and Anthony W.-C. Lun)
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Sun Tsu Suan Ching: $72$