A tetrahedron has faces that are congruent triangles with sides $a, b, c$. Express its volume in terms of the edges $a, b, c$!!
Suppose the tetrahedron exists. It is known that the faces of a tetrahedron are congruent if and only if the circumscribed parallelepiped is a rectangular parallelepiped. It is also a known theorem that if the faces of the tetrahedron are congruent, then the faces are acute triangles, and the volume of the tetrahedron is one-third of the volume of the circumscribed parallelepiped. These theorems (with proof guidance) can be found in the "Collection of Geometric Problems I," as problems 2088-2090 and 2083. Since the circumscribed parallelepiped is a rectangular parallelepiped, the opposite edges of the tetrahedron will be equal. Our diagram is thus constructed, based on which we determine the lengths of the edges of the parallelepiped:

\[
\begin{aligned}
& x^{2}+y^{2}=c^{2} \\
& y^{2}+z^{2}=b^{2} \\
& z^{2}+x^{2}=a^{2}
\end{aligned}
\]

From (1), \(x^{2}+y^{2}+z^{2}=\frac{a^{2}+b^{2}+c^{2}}{2}\), substituting the right-hand sides of the equations in (1):

\[
x=\sqrt{\frac{a^{2}-b^{2}+c^{2}}{2}}, \quad y=\sqrt{\frac{-a^{2}+b^{2}+c^{2}}{2}}, \quad z=\sqrt{\frac{a^{2}+b^{2}-c^{2}}{2}}
\]

The volume of the parallelepiped is \(x \cdot y \cdot z\), so the volume of the tetrahedron is:

\[
V=\frac{1}{3} x \cdot y \cdot z=\frac{1}{6 \cdot \sqrt{2}} \cdot \sqrt{\left(-a^{2}+b^{2}+c^{2}\right) \cdot\left(a^{2}-b^{2}+c^{2}\right) \cdot\left(a^{2}+b^{2}-c^{2}\right)}
\]

Since the faces of the tetrahedron are acute triangles,

\[
a^{2}<b^{2}+c^{2} \quad \text { and } \quad b^{2}<a^{2}+c^{2} \quad \text { and } \quad c^{2}<a^{2}+b^{2}
\]

thus the factors under the square root are positive.

Based on the work of Attila Maróti (Szeged, Ságvári E. Gymnasium, 4th grade)

Note. According to the problem statement, it was natural to assume that the tetrahedron exists, and in this case, condition (2) holds. Conversely, it can be shown that if (2) holds, then there exists a tetrahedron whose faces are congruent triangles with sides \(a, b, c\), and there is only one such tetrahedron.

![](https://cdn.mathpix.com/cropped/2024_05_02_a44c654ada95275e76c3g-1.jpg?height=356&width=593&top_left_y=1535&top_left_x=755)