In a certain right-angled triangle, one of the legs is in geometric mean proportion to the other two sides. Calculate its angles, and construct the triangle if the hypotenuse is given.
Let the hypotenuse of the right-angled triangle be $c$, and the two legs be $b > a$.

According to the problem,

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

By Pythagoras' theorem,

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

and thus

$$
c^{2}=a^{2}+a c=a(a+c)
$$

For the construction of the side $a$, see the solution of Exercise 186. To calculate the angles, express $a$ in terms of $c$ from (1). Using the quadratic formula,

$$
a=\frac{-c+\sqrt{c^{2}+4 c^{2}}}{2}=c \frac{\sqrt{5}-1}{2}
$$

and thus

$$
\sin \alpha=\frac{a}{c}=\frac{\sqrt{5}-1}{2}, \approx 0.6180
$$

from which

$$
\alpha=38^{\circ} 10^{\prime}, \quad \beta=51^{\circ} 50^{\prime}
$$

Bartha Gyöngyi (Bp., VIII., Apáczai Csere g. I. o. t.)