
2. Find all possible values of the quotient

$$
\frac{r+\rho}{a+b}
$$

where $r$ and $\rho$ are respectively the radii of the circumcircle and incircle of the right triangle with legs $a$ and $b$.


Solution. The distances between the vertices and points of tangency of the incircle (denoted according to the figure) of a triangle $A B C$ are

$$
|A U|=|A V|=\frac{b+c-a}{2}, \quad|B V|=|B T|=\frac{a+c-b}{2}, \quad|C T|=|C U|=\frac{a+b-c}{2},
$$

which is easy to obtain from

$$
|A V|+|B V|=c, \quad|A U|+|C U|=b, \quad|B T|+|C T|=a .
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_10b455106c778bb0ba10g-09.jpg?height=320&width=404&top_left_y=1436&top_left_x=498)

Fig. 1

The points $C, T, U$, and $S$ (the center of the incircle) form a deltoid, actually a square, if the angle $A C B$ is right. The side of the square is $\rho=|S U|=|C U|$. This gives

$$
\rho=\frac{a+b-c}{2}
$$

moreover according to the Thales theorem $r=\frac{1}{2} c$ and we get

$$
r+\rho=\frac{c}{2}+\frac{a+b-c}{2}=\frac{a+b}{2} .
$$

Conclusion. There is only one possible value of the quotient $(r+\rho) /(a+b)$ in any right triangle, namely $\frac{1}{2}$.
