In a certain right-angled triangle, the hypotenuse $c=12.4 \mathrm{~cm}$, and the angle bisector of $\beta$ is $f_{b}=8.7 \mathrm{~cm}$. What are the lengths of the legs?
Let the sought legs be denoted by $a$ and $b$, and for simplicity, write $f$ instead of $f_{b}$.

According to a known relationship,

$$
\cos \beta=2 \cos ^{2} \frac{\beta}{2}-1, \text { but } \cos \beta=\frac{a}{c} \text { and } \cos \frac{\beta}{2}=\frac{a}{f}
$$

and thus

$$
\frac{a}{c}=\frac{2 a^{2}}{f^{2}}-1
$$

which is a quadratic equation in $a$.

Rearranging,

$$
2 c a^{2}-f^{2} a-f^{2} c=0
$$

from which

$$
a=\frac{f^{2} \pm \sqrt{f^{4}+8 f^{2} c^{2}}}{4 c}=\frac{f^{2} \pm f \sqrt{f^{2}+8 c^{2}}}{4 c}
$$

Since the negative root has no meaning, therefore

$$
\begin{gathered}
a=\frac{8.7^{2}+8.7 \sqrt{8.7^{2}+8 \cdot 12.4^{2}}}{4 \cdot 12.4}=\frac{75.69+8.7 \sqrt{1305.77}}{49.6} \approx 7.865 \text{ cm} \\
b=\sqrt{12.4^{2}-7.865^{2}}=\sqrt{20.265 \cdot 4.535} \approx 9.590 \text{ cm}
\end{gathered}
$$

Pál Bártfai (Bp. IX. Ref. g. I. o. t.)