[ Mean proportionals in a right triangle ]

On the hypotenuse $B C$ of the right triangle $A B C$, there is a point $D$ such that $A D \perp B C$. Find the hypotenuse $B C$ if it is known that $A D=D C-B D=h$.

#
By the property of the height of a right triangle drawn from the vertex of the right angle, $A D^2 = C D \cdot B D$.

Let $C D = x, B D = y$. From the system

$$
\left\{\begin{array}{l}
x y = h^2 \\
x - y = h
\end{array}\right.
$$

we find that $x = \frac{h(1+\sqrt{5})}{2}, y = \frac{h(\sqrt{5}-1)}{2}$. Therefore,

$$
B C = x + y = \frac{h(1+\sqrt{5})}{2} + \frac{h(\sqrt{5}-1)}{2} = h \sqrt{5} .
$$

## Answer

$h \sqrt{5}$