Haidao Suanjing/Examples/Example 1

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: $72$