# Chiu Chang Suann Jing/Examples/Example 9

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## Example of Problem from

## Example of Problem from *Chiu Chang Suann Jing*

*What is the largest circle that can be inscribed within a right-angled triangle,**the two short sides of which are respectively $8$ and $15$?*

## Solution

The diameter of the circle is $6$.

## Proof

Let the lengths of the short sides be denoted $a$ and $b$.

Let the length of the hypotenuse be denoted $c$.

Without loss of generality let $a \le b$.

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 $2$ copies of this rectangle be assembled into one long rectangle whose length is $a + b + c$ and whose width is $D$.

Note that $D$ is the diameter of the circle that we are required to find.

Thus we have that:

- $D \paren {a + b + c} = 2 a b$

Setting $a = 8$, $b = 15$, from Pythagoras's Theorem we have that $c = 17$.

Thus we have that:

\(\ds D\) | \(=\) | \(\ds \dfrac {2 a b} {a + b + c}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {2 \times 8 \times 15} {8 + 15 + 17}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac {240} {40}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 6\) |

$\blacksquare$

## Sources

- c. 100: Anonymous:
*Chiu Chang Suann Jing* - 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:
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