34. A spherical triangle is the intersection of a sphere and a trihedral angle with its vertex at the center of the sphere (or a part of the sphere bounded by three arcs of great circles) (Fig. 6). Calculate the area of the spherical triangle \(ABC\), considering it as the intersection of three dihedral angles: one with vertex \(A\) and the corresponding angle of magnitude \(\alpha\), another with vertex \(B\) and the angle of magnitude \(\beta\), and the third with vertex \(C\) and the angle of magnitude \(\gamma\).

35*. Prove that if the faces of a tetrahedron are equal in area (i.e., have equal areas), then the circumscribed and inscribed spheres of the tetrahedron are concentric.

![](https://cdn.mathpix.com/cropped/2024_05_21_5cdadc99c18e78f1502ag-050.jpg?height=663&width=626&top_left_y=1710&top_left_x=1092)

Fig. 6
34. Figure 6 shows a hemisphere, the area of the surface of which is $2 \pi R^{2}$. This hemisphere is divided into three spherical digons with areas $S_{\alpha}, S_{\beta}, S_{\gamma}$. Here, the digons $S_{\alpha}$ and $S_{\beta}$ are entirely contained within the hemisphere, while the digon $S_{\gamma}$ is "partially": by the spherical triangle $A B C$ and an additional part that is centrally symmetric to it, congruent with the "missing" part of $S_{\gamma}$. The triangle $A B C$ is contained in each of these digons. Therefore, to obtain the area of the hemisphere's surface, one can subtract twice the area of triangle $A B C$ from the sum of the areas of the three spherical digons, leading to:

$$
\begin{gathered}
2 \pi R^{2}=2 R^{2} \alpha+2 R^{2} \beta+2 R^{2} \gamma-2 S_{\Delta} \\
S_{\Delta}=(\alpha+\beta+\gamma-\pi) R^{2}
\end{gathered}
$$

Note that $\alpha+\beta+\gamma$ is the sum of the dihedral angles of the trihedral angle $O A B C$, which is always greater than $\pi$, corresponding to the positivity of the area of the spherical triangle. The difference $\sigma=\alpha+\beta+\gamma-\pi$ is called the spherical excess of the triangle. The formula for the area of a spherical triangle can be written in terms of the excess as: $S_{\Delta}=\sigma R^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_5cdadc99c18e78f1502ag-191.jpg?height=528&width=654&top_left_y=250&top_left_x=358)

Fig. 115