Prokop constructed a triangle $ABC$, where the interior angle at vertex $A$ was greater than $60^{\circ}$ and the interior angle at vertex $B$ was less than $60^{\circ}$. Jirka drew a point $D$ in the half-plane defined by the line $AB$ and point $C$, such that triangle $ABD$ was equilateral. Then, the boys found that triangles $ACD$ and $BCD$ were isosceles with the main vertex at $D$.

Determine the measure of angle $ACB$.

(E. Semerádová)

Hint. Find the relationships between the interior angles of the mentioned triangles.
The sizes of the internal angles in triangle $ABC$ are denoted as $\alpha, \beta, \gamma$ respectively. In the equilateral triangle $ABD$, all internal angles have a size of $60^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_c15208fdb3e01b0f90f8g-4.jpg?height=534&width=574&top_left_y=184&top_left_x=747)

The equal angles at the base of the isosceles triangle $BCD$ have a size of

$$
|\Varangle BCD| = |\Varangle CBD| = |\Varangle ABD| - |\Varangle ABC| = 60^{\circ} - \beta.
$$

The equal angles at the base of the isosceles triangle $ACD$ have a size of

$$
|\Varangle ACD| = |\Varangle CAD| = |\Varangle CAB| - |\Varangle DAB| = \alpha - 60^{\circ}.
$$

The size of the unknown angle $ACB$ can be expressed as

$$
\gamma = |\Varangle ACD| - |\Varangle BCD| = (\alpha - 60^{\circ}) - (60^{\circ} - \beta) = \alpha + \beta - 120^{\circ}.
$$

The sum of the sizes of the internal angles in triangle $ABC$ is $180^{\circ}$, thus

$$
\alpha + \beta + (\alpha + \beta - 120^{\circ}) = 180^{\circ},
$$

from which it follows that $\alpha + \beta = 150^{\circ}$. The angle $ACB$ has a size of $\gamma = 150^{\circ} - 120^{\circ} = 30^{\circ}$.

Note. The given conditions correspond to infinitely many situations; $\gamma$ is always $30^{\circ}$, and the remaining $150^{\circ}$ can be distributed arbitrarily between $\alpha$ and $\beta$.

All points $A, B, C$ lie on a circle with center at point $D$. In such cases, it generally holds that the size of the angle $ACB$ is half the size of the angle $ADB$ (see the theorem about the inscribed and central angles).