$C$ is the midpoint of segment $A B$, $P$ is a point on the circle with diameter $A B$, and $Q$ is a point on the circle with diameter $A C$. What is the relationship between the lengths of segments $A P$ and $A Q$ if $P Q$ is perpendicular to $A B$?
Let the intersection point of lines $P Q$ and $A B$ be $T$. The triangles $A Q C$ and $A P B$ are right-angled according to Thales' theorem, and $T$ is the foot of the altitude to the hypotenuse in these triangles.

![](https://cdn.mathpix.com/cropped/2024_05_02_f5f0947ffa6fe86e0df5g-1.jpg?height=512&width=545&top_left_y=270&top_left_x=779)

Thus, according to the leg theorem:

\[
\begin{array}{r}
A Q^{2}=A T \cdot A C \\
A P^{2}=A T \cdot A B=2 A T \cdot A C
\end{array}
\]

since $C$ is the midpoint of segment $A B$, and therefore $A B=2 A C$; hence $A P^{2}=2 A Q^{2}$.

Therefore, the length of segment $A P$ is $\sqrt{2}$ times the length of segment $A Q$.

Szeidl Ádám (Miskolc, Földes F. Gimn., I. o. t.)