B2. Points $A, B, C$ and $D$, which divide the circle in the ratio $3: 5: 7: 3$, determine a cyclic quadrilateral. Draw a sketch and calculate the interior angles of the resulting quadrilateral.
B2. The vertices of a cyclic quadrilateral $ABCD$ divide the circle in the ratio 3:5:7:3. The segments connecting the center of the circle $S$ to these points divide the full angle in the same ratio. Thus, $\angle ASB = 60^\circ$, $\angle BSC = 100^\circ$, $\angle CSD = 140^\circ$, and $\angle DSA = 60^\circ$. Now we can quickly find the sizes of the interior angles of the quadrilateral, as triangles $ABS$ and $DAS$ are equilateral, and triangles $BCS$ and $CDS$ are isosceles with angles of $40^\circ$ and $20^\circ$ at the base. The interior angles are therefore $120^\circ$, $100^\circ$, $60^\circ$, and $80^\circ$.

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-12.jpg?height=440&width=483&top_left_y=1022&top_left_x=1392)

We can solve it differently: from the given ratio, we conclude that the individual arcs are $3t$, $5t$, $7t$, and $3t$. The inscribed angles at $A$, $B$, $C$, and $D$ correspond to the arcs $5t + 7t = 12t$, $7t + 3t = 10t$, $3t + 3t = 6t$, and $3t + 5t = 8t$, respectively, so their sizes are in the ratio 12:10:6:8 or 6:5:3:4. Since the sum of the interior angles of the quadrilateral is $360^\circ$, we have $6v + 5v + 3v + 4v = 360^\circ$, from which we get $v = 20^\circ$ and finally the interior angles: $120^\circ$, $100^\circ$, $60^\circ$, and $80^\circ$.

Appropriate sketch ...................................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-12.jpg?height=51&width=1641&top_left_y=1822&top_left_x=276)

Conclusion: $6x + 5x + 3x + 4x = 360^\circ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-12.jpg?height=51&width=1639&top_left_y=1919&top_left_x=274)

![](https://cdn.mathpix.com/cropped/2024_06_07_4401d59a37898c9fd2b4g-12.jpg?height=54&width=1637&top_left_y=1966&top_left_x=278)