12.025. In a right triangle $ABC$, the acute angle $A$ is equal to $\alpha$ radians. An arc of a circle centered at the right angle vertex $C$ touches the hypotenuse at point $D$ and intersects the legs $AC$ and $BC$ at points $E$ and $F$, respectively. Find the ratio of the areas of the curvilinear triangles $ADE$ and $BDF$.
## Solution.

By the condition in $\triangle A C B \quad \angle A C B=90^{\circ}, \angle C A B=\alpha$ (Fig. 12.28). Let $A C=x$. From $\triangle A D C, \angle A D C=90^{\circ}, C D=x \sin \alpha$, then

$$
\begin{aligned}
& \left.S_{C E D}=\frac{1}{2} x^{2} \sin ^{2} \alpha \cdot\left(\frac{\pi}{2}-\alpha\right) \text { (i.e. } \angle A C D=\frac{\pi}{2}-\alpha\right) ; S_{\triangle A D C}=\frac{C D \cdot A D}{2}= \\
& =\frac{x^{2} \cos \alpha \sin \alpha}{2} ; \quad S_{A D E}=S_{\triangle A D C}-S_{C E D}=\frac{x^{2} \cos \alpha \sin \alpha}{2}-\frac{x^{2} \sin ^{2} \alpha\left(\frac{\pi}{2}-\alpha\right)}{2}
\end{aligned}
$$

From $\triangle C D B, \angle C D B=90^{\circ}, D B=x \sin \alpha \cdot \operatorname{tg} \alpha, S_{\triangle C D B}=\frac{x^{2}}{2} \sin ^{2} \alpha \cdot \operatorname{tg} \alpha$,

$$
\begin{aligned}
& S_{C D F}=\frac{1}{2} x^{2} \cdot \sin ^{2} \alpha \cdot \alpha, S_{D F B}=S_{\triangle C D B}-S_{C D F}=\frac{x^{2} \sin ^{2} \alpha \operatorname{tg} \alpha}{2}-\frac{x^{2} \sin ^{2} \alpha \cdot \alpha}{2} \\
& \frac{S_{D F B}}{S_{A E D}}=\frac{\sin ^{2} \alpha(\operatorname{tg} \alpha-\alpha)}{\sin ^{2} \alpha\left(\operatorname{ctg} \alpha-\frac{\pi}{2}+\alpha\right)}=\frac{\operatorname{tg} \alpha-\alpha}{\operatorname{ctg} \alpha-\frac{\pi}{2}+\alpha} \\
& \quad \text { Answer: } \frac{\operatorname{tg} \alpha-\alpha}{\operatorname{ctg} \alpha-\frac{\pi}{2}+\alpha}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-734.jpg?height=316&width=496&top_left_y=114&top_left_x=128)

Fig. 12.28

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-734.jpg?height=340&width=582&top_left_y=94&top_left_x=649)

Fig. 12.29