Let's construct a right-angled triangle, given the hypotenuse $a$ and that the difference between the legs is equal to the height corresponding to the hypotenuse.
If we denote the legs by $b$ and $c$, and the height corresponding to the hypotenuse by $m$, then according to the condition:

$$
b^{2}+c^{2}=a^{2}
$$

$$
m^{2}=(b-c)^{2}=b^{2}+c^{2}-2 b c
$$

and

$$
b c=a m
$$

This is a system of three equations with three unknowns, from which $m$ can be calculated. Indeed:

$$
(1)+(2) \quad a^{2}-m^{2}=2 b c
$$

or, considering (3):

$$
m^{2}+2 a m-a^{2}=0
$$

thus

$$
m=-a+a \sqrt{2}=a \sqrt{2}-a .
$$

Only the positive value of $\sqrt{2}$ is valid, as $m$ cannot be negative. According to this, the construction proceeds as follows:

If we draw a right triangle $ABD$ where $AB=BD=a$, then $AD=a \sqrt{2}$. If we now cut off $DE=a$ from $DA$, then:

$$
m=AE=a \sqrt{2}-a
$$

Knowing the hypotenuse and the corresponding height, the desired right triangle can be easily constructed.

(Pilczer Pál, Kaposvár.)

The problem was also solved by: Aczél F., Bartók I., Bayer B., Bogdán G., Dessauer A., Haar A., Hirchfeld Gy., Kertész F., König D., Lázár L., Póka Gy., Riesz K., Sasvári J., Sümegi Gy., Smodics H., Tóbiás J. L, Wohlstein S.