135. Inside a regular triangular pyramid, there is a vertex of a trihedral angle, all plane angles of which are right angles, and the bisectors of the plane angles pass through the vertices of the base. In what ratio does the surface of this angle divide the volume of the pyramid, if each face of the pyramid is divided by it into two equal parts?
135. In Fig. 21, \( SABC \) is a given pyramid, \( SO \) is its height, and \( G \) is the vertex of the trihedral angle. From the condition, it follows that \( G \) is located on \( SO \). Moreover, the faces of the trihedral angle, when intersecting the plane of the base \( ABC \), form an equilateral triangle, the sides of which are parallel to the sides of \( \triangle ABC \) and pass through its vertices. Therefore, if one of the edges of the trihedral angle intersects the plane \( ABC \) at point \( E \), and the face \( CSB \) at point \( F \), then \( F \) lies on the apothem \( SD \) of the lateral face \( CSB \) and \( |ED| = |DA| \).

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-079.jpg?height=373&width=790&top_left_y=1256&top_left_x=225)

Fig. 21.

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-080.jpg?height=461&width=400&top_left_y=255&top_left_x=196)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-080.jpg?height=461&width=386&top_left_y=791&top_left_x=199)
b)

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-080.jpg?height=463&width=388&top_left_y=258&top_left_x=682)

b)

![](https://cdn.mathpix.com/cropped/2024_05_21_9b03e22d2b8ca67aa653g-080.jpg?height=461&width=388&top_left_y=795&top_left_x=671)

g)

Fig. 22.

According to the condition, \( |SF| = |FD| \). Draw a line through \( S \) parallel to \( EO \), and denote by \( K \) the point of intersection of this line with the line \( EF \) (Fig. 21, b). We have \( |SK| = |ED| \). Therefore, \( \frac{|SG|}{|GO|} = \frac{|SK|}{|EO|} = \frac{|ED|}{|EO|} = \frac{3}{4} \). Thus, the volume of the pyramid \( GABC \) is \( \frac{4}{7} \) of the volume of the pyramid \( SABC \).

On the other hand, the constructed trihedral angle divides the part of the pyramid \( SABC \) located above the pyramid \( GABC \) into two equal parts.

Answer. The volume of the part of the pyramid located outside the trihedral angle to the volume of the part inside it is \( 3:11 \). 136. \( \frac{V}{6} \).