Mekkora a befogók aránya abban a derékszögú háromszögben, amelynek magasságaiból mint oldalakból szerkesztett háromszög szintén derékszögú?

What is the ratio of the legs in a right triangle where the heights drawn to the legs form a right triangle when used as sides?
Let the legs of the right triangle be denoted by $a$ and $b$ $(a \leq b)$, and the height corresponding to the hypotenuse by $m$.

The area of the triangle can be written in two ways:

$$
\frac{a b}{2}=\frac{c m}{2}
$$

After squaring and rearranging, we get $m^{2}=\frac{a^{2} b^{2}}{a^{2}+b^{2}}$. Clearly, $m<a$, and thus we can apply the Pythagorean theorem to the heights:

$$
a^{2}+\frac{a^{2} b^{2}}{a^{2}+b^{2}}=b^{2}
$$

Performing the operations and rearranging:

$$
a^{4}-b^{4}+a^{2} b^{2}=0
$$

Dividing the equation by $a^{2} b^{2} \neq 0$, we get:

$$
\frac{a^{2}}{b^{2}}-\frac{b^{2}}{a^{2}}+1=0
$$

By introducing the new variable $\frac{a^{2}}{b^{2}}=x$, we arrive at the quadratic equation $x-\frac{1}{x}+1=0$, whose roots are: $x_{1,2}=\frac{-1 \pm \sqrt{5}}{2}$. Only the positive root is valid, thus

$$
\frac{a}{b}=\sqrt{\frac{-1+\sqrt{5}}{2}} \approx 0.786
$$

George Szentzi (Székesfehérvár, 323. Vörösmarty M. Ind. High School, 2nd grade)