## Task $2 / 84$

Given is a circle with diameter $d=2 r=A B$. A line perpendicular to $A B$ intersects the diameter at $P$ and the circle at $C$ and $D$. The perimeters of triangles $A P C$ and $B P D$ are in the ratio $\sqrt{3}: 1$.

What is the ratio $A P: P B$?
If the perimeters of the similar triangles behave like $\sqrt{3}: 1$, then the areas behave like $3: 1$. Consequently, it holds that

$$
0.5 \cdot A P \cdot P C: 0.5 \cdot B P \cdot P D=3: 1
$$

Due to $\mathrm{PC}=\mathrm{PD}$ (for symmetry reasons), it immediately follows that $A P: P B=3: 1$.