3. Construct a right triangle $ABC$, with the right angle at vertex $C$, if the hypotenuse $c$ is given, and the median $t_{c}$ is the geometric mean of the legs.
Solution. By the condition we have $t_{c}^{2}=a b ;$ but, $t_{c}=\frac{c}{2}$, so, it means, $c^{2}=4 a b$. For the height $h_{c}$ we have $h_{c}=\frac{a b}{c}=\frac{c}{4}$, so the triangle $A B C$ (with given $c$ and $h_{c}$) can be constructed.

Construction. 1) $\overline{A B}=c$;
2) $O$ is the midpoint of $A B$;

3) line $p$, parallel to $A B$ and at a distance $h_{c}=\frac{c}{4}$ from it;
4) circle $k(O, \overline{O A})$;
5) vertex $C$ lies on $P \cap k$ (see diagram).

![](https://cdn.mathpix.com/cropped/2024_06_05_a33abaf1f90f1a4be6bag-4.jpg?height=332&width=377&top_left_y=471&top_left_x=1085)