A triangle's two altitudes are at least as long as the sides to which they are drawn. What are the angles of the triangle?

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I. solution. The area of the triangle, with the usual notation, is $t=\frac{a \cdot b \cdot \sin \gamma}{2}=\frac{a \cdot m_{a}}{2}=\frac{b \cdot m_{b}}{2}$. Since $m_{a} \geq a$ and $m_{b} \geq b, t \geq \frac{a^{2}}{2}$ and $t \geq \frac{b^{2}}{2}$. Therefore, $\frac{a \cdot b \cdot \sin \gamma}{2} \geq \frac{a^{2}}{2}$ and $\frac{a \cdot b \cdot \sin \gamma}{2} \geq \frac{b^{2}}{2}$. Since $a, b$ are positive numbers, we get that $\sin \gamma \geq \frac{a}{b}$ and $\sin \gamma \geq \frac{b}{a}$. From these, it follows that $\sin \gamma \geq 1$, which is only possible if $\sin \gamma=1$, i.e., $\gamma=90^{\circ}$.

In a right-angled triangle, the height corresponding to one leg is the other leg, so according to the conditions, $b \geq a$ and $a \geq b$, from which $a=b$ follows.

Thus, the triangle is right-angled and isosceles, so its angles are: $90^{\circ}, 45^{\circ}, 45^{\circ}$.

Zalán Péter (Budapesti Evangélikus Gimn., 9. o.t.)

II. solution. In our diagram, the two heights about which the two conditions speak are $m_{a}$ and $m_{b}$. According to the conditions, $m_{a} \geq a$ and $m_{b} \geq b$. Let's apply the Pythagorean theorem to the triangles $A N C$ and $B M C$, and use the conditions:

$$
\begin{aligned}
m_{a}^{2}+N C^{2} & =b^{2} \leq m_{b}^{2} \\
m_{b}^{2}+M C^{2} & =a^{2} \leq m_{a}^{2}
\end{aligned}
$$

These relationships are also valid for degenerate triangles $A N C$ and $B M C$. Add the obtained inequalities:

$$
m_{a}^{2}+N C^{2}+m_{b}^{2}+M C^{2} \leq m_{b}^{2}+m_{a}^{2}
$$

from which it follows: $N C^{2}+M C^{2} \leq 0$. This is only possible if $N C=M C=0$, and thus $\gamma=90^{\circ}$, and $m_{a}=b$ and $m_{b}=a$. Therefore, considering the conditions, $b \geq a$ and $a \geq b$, from which $a=b$. The triangle is thus right-angled and isosceles, so its angles are: $90^{\circ}, 45^{\circ}, 45^{\circ}$.

Breuer János (Budapest, ELTE Apáczai Csere J. Gimn., 9. o.t.)

![](https://cdn.mathpix.com/cropped/2024_05_02_83d032f95bf3ab5c2a7ag-1.jpg?height=434&width=539&top_left_y=1130&top_left_x=782)