Find the angles of a triangle if it is known that all squares inscribed in it are equal (each of the squares is inscribed such that two of its vertices lie on one side of the triangle, and the other vertices on the other two sides of the triangle).

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Let the sides of the triangle be \(a, b\), and \(c\), and the heights dropped to them be \(h_{a}, h_{b}\), and \(h_{c}\). According to problem 53756, the sides of the inscribed squares are \(\frac{a h_{a}}{a + h_{a}}, \frac{b h_{b}}{b + h_{b}}, \frac{c h_{c}}{c + h_{c}}\). By the condition, these numbers are equal.

The numerators of the obtained fractions are equal to each other (this is twice the area of the triangle). Therefore, the denominators are also equal:

\[a + h_{a} = b + h_{b} = c + h_{c}.\] 

Since \(a h_{a} = b h_{b} = c h_{c}\), the three pairs of numbers expressing the side and the height dropped to it of the given triangle must satisfy the same quadratic equation. Due to the uniqueness of the pair of solutions of a quadratic equation, these three pairs coincide, which implies that triangle \(ABC\) is equilateral (considering that the smallest height is less than the smallest side).

## Answer

All angles are \(60^{\circ}\).