12.60*. In an acute-angled triangle $A B C$, the segments $B O$ and $C O$, where $O-$ is the center of the circumscribed circle, are extended to intersect the sides $A C$ and $A B$ at points $D$ and $E$. It turns out that $\angle B D E=50^{\circ}$ and $\angle C E D=30^{\circ}$. Find the measures of the angles of triangle $A B C$.

## §8. Circles
12.60. Since $\angle B D E=50^{\circ}$ and $\angle C E D=30^{\circ}$, then $\angle B O C=\angle E O D=$ $=180^{\circ}-50^{\circ}-30^{\circ}=100^{\circ}$. Let's assume that the diameters $B B^{\prime}$ and $C C^{\prime}$ of the circle are fixed, with $\angle B O C=100^{\circ}$, and point $A$ moves along the arc $B^{\prime} C^{\prime}$. Let $D$ be the intersection of $B B^{\prime}$ and $A C$, and $E$ be the intersection of $C C^{\prime}$ and $A B$ (Fig. 12.6). As point $A$ moves from $B^{\prime}$ to $C^{\prime}$, segment $O E$ increases, while $O D$ decreases, so angle $O E D$ decreases, and angle $O D E$ increases. Therefore, there is a unique position of point $A$ where $\angle C E D=\angle O E D=30^{\circ}$ and $\angle B D E=\angle O D E=$ $=50^{\circ}$.

Now let's prove that the triangle $A B C$ with angles $\angle A=50^{\circ}, \angle B=70^{\circ}, \angle C=60^{\circ}$ has the required property. Let $A_{1} \ldots A_{18}$ be a regular 18-sided polygon. We can take the triangle $A B C$ as the triangle $A_{2} A_{14} A_{9}$. The diagonal $A_{1} A_{12}$ passes through point $E$ (see the solution to problem 12.58). Let $F$ be the intersection of the lines $A_{1} A_{12}$ and $A_{5} A_{14}$; the line $A_{9} A_{16}$ is symmetric to the line $A_{1} A_{12}$ with respect to the line $A_{5} A_{14}$, so it passes through point $F$. In triangle $C D F$, ray $C E$ is the angle bisector of angle $C$, and line $F E$ is the bisector of the external angle at vertex $F$. Therefore, $D E$ is the angle bisector of angle $A D B$, i.e., $\angle O D E=\left(\smile A_{2} A_{14}+\smile A_{5} A_{9}\right) / 4=50^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_22b0ba5e8c5a1f49bd81g-285.jpg?height=417&width=382&top_left_y=819&top_left_x=1183)

Fig. 12.6