## Task 3 - 110923

A circle is divided into 30 equal arcs. The points of division are sequentially labeled as $P_{1}$ to $P_{30}$.

Calculate the size of each of the four angles at which the segments $P_{7} P_{18}$ and $P_{12} P_{21}$ intersect!
We supplement the 30-gon to a 60-gon with vertices $Q_{1}, \ldots, Q_{60}$. Let M be the center.

Then $P_{7} P_{18}=Q_{14} Q_{36}$ is orthogonal to $M Q_{25}, 25=\frac{1}{2}(14+36)$ and $P_{12} P_{21}=Q_{24} Q_{42}$ is orthogonal to $M Q_{33}, 33=\frac{1}{2}(24+42)$.

The angle of intersection of the lines therefore corresponds to the angle $\varangle Q_{25} M Q_{33}=\frac{360^{\circ}}{60}(33-25)=48^{\circ}$.