9. (20 points) On the sides of a triangle, isosceles triangles are constructed outward with angles at the vertices $\alpha, \beta$ and $\gamma$, where $\alpha+\beta+\gamma=360^{\circ}$. Find the angles of the triangle formed by the vertices of these isosceles triangles.

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# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_9c64a952bd96df941010g-09.jpg?height=848&width=1334&top_left_y=378&top_left_x=378)

Since the sum of the interior angles of a hexagon is $720^{\circ}$, then

$$
\begin{gathered}
\angle Z A Y + \angle Y C X + \angle X B Z = 720^{\circ} - (\angle A Y C + \angle C X B + \angle B Z A) = \\
= 720^{\circ} - (\alpha + \beta + \gamma) = 720^{\circ} - 360^{\circ} = 360^{\circ}
\end{gathered}
$$

Perform a rotation around point $X$ by $\angle C X B$ counterclockwise. Point $C$ will move to point $B$, and point $Y$ will move to point $Y^{\prime}$. Therefore, $\triangle X C Y$ will be equal to $\triangle X B Y^{\prime}$. From this, we get that $X Y = X Y^{\prime}$ and $\angle X Y^{\prime} B = \angle X Y C$.

Now consider $\triangle Z B Y^{\prime}$:

$$
\angle Z B Y^{\prime} = 360^{\circ} - \angle Y^{\prime} B X - \angle X B Z = 360^{\circ} - \angle Y C X - \angle X B Z = \angle Z A Y
$$

We get that $\triangle Z B Y^{\prime} = \triangle Z A Y$ by two sides and the included angle. Therefore, $Z Y^{\prime} = Z Y$ and $\angle B Y^{\prime} Z = \angle A Y Z$. It remains to note that $\triangle Y^{\prime} Z X = \triangle Y Z X$ by three sides, so $\angle Z Y^{\prime} X = \angle Z Y X$.

In the end, we get

$$
\begin{gathered}
\angle A Y C = \angle A Y Z + \angle Z Y X + \angle X Y C = \angle B Y^{\prime} Z + \angle Z Y X + \angle X Y^{\prime} B = \\
= \angle Z Y^{\prime} X + \angle Z Y X = 2 \angle Z Y X
\end{gathered}
$$

Therefore, $\angle Z Y X = \frac{1}{2} \angle A Y C = \frac{\alpha}{2}$. Similarly, it can be shown that $\angle Y X Z = \frac{\beta}{2}$ and $\angle X Z Y = \frac{\gamma}{2}$.

Answers:

| Option | 1 | 2 | 3 | 4 | 5 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $50^{\circ}, 60^{\circ}, 70^{\circ}$ | $50^{\circ}, 55^{\circ}, 75^{\circ}$ | $55^{\circ}, 60^{\circ}, 65^{\circ}$ | $40^{\circ}, 65^{\circ}, 75^{\circ}$ | $45^{\circ}, 65^{\circ}, 70^{\circ}$ |
| 6 | 7 | 8 | 9 | 10 | 11 |
| $40^{\circ}, 60^{\circ}, 80^{\circ}$ | $45^{\circ}, 50^{\circ}, 75^{\circ}$ | $45^{\circ}, 50^{\circ}, 85^{\circ}$ | $50^{\circ}, 50^{\circ}, 80^{\circ}$ | $40^{\circ}, 70^{\circ}, 70^{\circ}$ | $30^{\circ}, 75^{\circ}, 75^{\circ}$ |