3. How many spheres exist that touch all the faces of the given triangular pyramid $T$?


3. The task is to determine how many points (centers of the sought spheres) are equidistant from the four faces of the pyramid.

The geometric locus of points equidistant from the faces of a given dihedral angle is a plane passing through the edge of the dihedral angle and bisecting this angle — the bisector plane of the dihedral angle.

Since two intersecting planes form two pairs of vertical dihedral angles, the geometric locus of points equidistant from two intersecting planes consists of two planes passing through the line of intersection of the first two planes.

The geometric locus of points equidistant from the faces of a trihedral angle is the line of intersection of the bisector planes of the three dihedral angles of the trihedral angle — the bisector line of the trihedral angle.

Since three intersecting planes form four pairs of vertical trihedral angles, the geometric locus of points equidistant from three intersecting planes consists of four lines passing through the point of intersection of the three planes.

Let $\mathrm{II}_{1}, \mathrm{II}_{2}, \mathrm{II}_{3}$, and $\mathrm{II}_{4}$ be the planes of the faces of the pyramid. Consider the trihedral angle of the pyramid formed by the planes $\mathrm{II}_{1}, \mathrm{II}_{2}$, and $\mathrm{II}_{3}$. The geometric locus of points equidistant from the three faces of this angle consists of four lines $\beta_{1}, \beta_{2}, \beta_{3}$, and $\beta_{4}$. Further, the geometric locus of points equidistant from the planes $\mathrm{II}_{1}$ and $\mathrm{II}_{4}$ consists of two planes $B_{1}$ and $B_{2}$. It is clear that each of the points of intersection of one of the lines $\beta_{1}, \beta_{2}, \beta_{3}$, and $\beta_{4}$ with one of the planes $B_{1}$ and $B_{2}$ (and only these points) will be equidistant from the planes $\mathrm{II}_{1}, \mathrm{II}_{2}, \mathrm{II}_{3}$, and $\mathrm{II}_{4}$. Thus, in general, we obtain eight points equidistant from the planes of the faces of the pyramid, and consequently, eight spheres tangent to these faces. It is not difficult to see that out of these eight spheres, one is enclosed inside the pyramid (the inscribed sphere; Fig. 35, a), four are located outside the pyramid, one inside each of its trihedral angles (the "exscribed" spheres; Fig. 35, b), and three are located inside the dihedral angles of the pyramid and inside the angles vertical to the opposite dihedral angles (Fig. 35, c).

![](https://cdn.mathpix.com/cropped/2024_05_21_68334347b64402497ba5g-097.jpg?height=847&width=788&top_left_y=841&top_left_x=526)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_68334347b64402497ba5g-097.jpg?height=852&width=753&top_left_y=805&top_left_x=1359)
b)

![](https://cdn.mathpix.com/cropped/2024_05_21_68334347b64402497ba5g-097.jpg?height=845&width=986&top_left_y=1832&top_left_x=827)
c:

Fig. 35.

Note that in special cases, a tetrahedral pyramid may have fewer than eight spheres tangent to the planes of all its faces. For example, a regular tetrahedron has only five such spheres (spheres of only the types shown in Fig. 35, a and b). This is because in some cases, some of the lines $\beta_{1}, \beta_{2}, \beta_{3}$, and $\beta_{4}$ may be parallel to some of the planes $B_{1}$ and $B_{2}$.

Remark. It can be shown that if the sum of the areas of two faces of a triangular pyramid is equal to the sum of the areas of the other two faces, then there are only 7 spheres tangent to the planes of all the faces; if the areas of the faces are pairwise equal, then there are 6 such spheres; finally, if all the faces of the pyramid are equal in area (and therefore equal; see, for example, problem 29 in the book by D. O. Shklyarsky, N. N. Chentsov, and I. M. Yaglom, Selected Problems and Theorems of Elementary Mathematics, part 3, "Mathematical Circle Library," issue 3), then there are only 5 such spheres. See "Elementary Geometry" by J. Hadamard, part 2 (Moscow, Uchpedgiz, 1951), solution to problem 701.