Chiu Chang Suann Jing/Examples/Example 9

Example of Problem from

 * 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 $c$.

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.


 * Diameter-of-inscribed-circle-9-chapters.png

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: