Example 3. Connect the midpoints of the sides of an acute triangle with side lengths $2 a, 2 b, 2 c$, forming four triangles, which is the unfolded diagram of a tetrahedron. Find the volume of this tetrahedron.
Given the figure, let the tetrahedron be \(A_{1} B C_{1} D\). According to the problem, each face of the tetrahedron is an isosceles triangle, and the three pairs of opposite edges are equal in length. Therefore, it can be extended to form a rectangular prism \(A B C D A_{1} B_{1} C_{1} D_{1}\). At this point, each side of the tetrahedron's three edges become the diagonals of the three faces of the rectangular prism that share a common vertex. Let \(A B = x\), \(A D = y\), and \(A A_{1} = z\).

\[
\begin{aligned}
\because A_{1} D = a, A_{1} B = b, B D = c, \\
\therefore x^{2} + y^{2} = c^{2}, \\
y^{2} + z^{2} = a^{2}, \\
x^{2} + z^{2} = b^{2}.
\end{aligned}
\]

It is easy to get \(x^{2} = \frac{1}{2}\left(b^{2} + c^{2} - a^{2}\right)\),
\[
\begin{array}{l}
y^{2} = \frac{1}{2}\left(a^{2} + c^{2} - b^{2}\right) \\
z^{2} = \frac{1}{2}\left(a^{2} + b^{2} - c^{2}\right).
\end{array}
\]

Thus, \(V_{A_{1} B C_{1} D} = \frac{1}{3} V_{\text {rectangular prism}} = \frac{1}{3} \sqrt{x^{2} y^{2} z^{2}} \frac{\sqrt{2}}{12}\)
- \(\sqrt{\left(b^{2} + c^{2} - a^{2}\right)\left(a^{2} + c^{2} - b^{2}\right)\left(a^{2} + b^{2} - c^{2}\right)}\).