4. On the plane, there are points $A, B, C$ and $D$ such that $|A B|=|B C|=$ $|A C|=|C D|=10 \mathrm{~cm}$ and $|A D|=17 \mathrm{~cm}$. What is the measure of angle $\Varangle A D B$?
II/4. Since point $C$ is equally distant from points $A, B$, and $D$, points $A, B$, and $D$ lie on a circle $\mathrm{s}$ with center $C$ and radius $10 \mathrm{~cm}$. Because $|A B|=10 \mathrm{~cm}$, triangle $A B C$ is equilateral and $\angle A C B=60^{\circ}$. Since $|A D|>|A B|$, point $D$ lies on the same side of line $A B$ as point $C$. Therefore, $\angle A D B$ and $\angle A C B$ are inscribed and central angles over the same arc, and $\angle A D B=30^{\circ}$. (In the figure, one of the two possible positions of point $D$ is drawn.)

![](https://cdn.mathpix.com/cropped/2024_06_07_af92d6d33b11d7748e87g-08.jpg?height=434&width=417&top_left_y=1913&top_left_x=1413)

Cyclic nature of points $A, B, D: \mathbf{2}$ points. Central angle $60^{\circ}: \mathbf{1}$ point. Inscribed angle $30^{\circ}: \mathbf{1}$ point. Justification that because $|A D|>|A B|$, points $C$ and $D$ lie on the same side of line $A B$: 3 points.

If the contestant (incorrectly!) claims that the position of point $D$ is uniquely determined by the conditions of the problem: at most 6 points.