9.11. (ČSSR, 72). Find all pairs of positive numbers $a, b \in \mathbf{R}$, for which there exist a right triangle $C D E$ and points $A, B$ on its hypotenuse $D E$, satisfying the conditions $\overrightarrow{D A}=\overrightarrow{A B}=\overrightarrow{B E}$ and $A C=a$, $B C=b$.
9.11. Let $\overrightarrow{C A}=x, \overrightarrow{C B}=\boldsymbol{y}$, then (Fig. 7)

$$
\overrightarrow{D A}=\overrightarrow{B E}=\overrightarrow{A B}=y-x, \overrightarrow{C E}=2 y-x, \overrightarrow{C D}=2 x-y ;
$$

the condition $C D \perp C E$ is equivalent to the equality

$$
(2 x-y)(2 y-x)=0 \text {, or } 5(x-y)^{2}=x^{2}+y^{2} .
$$

The last condition means that the sides $B C=a, A C=b, A B=c$ of triangle $A B C$ satisfy the equality $5 c^{2}=a^{2}+b^{2}$. Such a
triangle exists for those and only those values of $a, b$ for which the triangle inequalities

$$
|a-b|<c<a+b
$$

are satisfied. Since $c>0$, the left inequality is equivalent to

![](https://cdn.mathpix.com/cropped/2024_05_21_03a501298610b72904feg-177.jpg?height=335&width=622&top_left_y=365&top_left_x=210)

Fig. 7

the condition $(2 a-b)(2 b-a)>0$. Thus, we obtain the answer: $1 / 2 < a / b < 2$.