16. Division of a Triangle. Divide a triangle into 19 triangles so that the same number of sides meet at each vertex of the resulting figure (as well as at the vertices of the large triangle).

In this problem, the number 19 cannot be replaced by a larger number, but it can be replaced by smaller numbers. Which ones?
16. To divide a triangle into a certain number of triangles so that the same number of sides meet at each vertex of the resulting figure, we can use regular polyhedra whose faces are triangles. These can be the following polyhedra: the regular tetrahedron, octahedron, and icosahedron, and only these.

If inside a tetrahedron we choose a point close to the center of one of its faces and project the edges of the tetrahedron onto a plane from this point, we will get the figure shown in Fig. 15. It consists of three triangles corresponding to the faces of the tetrahedron; the fourth face, when projected, turns into

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-071.jpg?height=368&width=368&top_left_y=1552&top_left_x=433)

Fig. 15.

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-071.jpg?height=317&width=368&top_left_y=1575&top_left_x=844)

Fig. 16.

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-071.jpg?height=317&width=368&top_left_y=1578&top_left_x=1255)

Fig. 17.

a large triangle \(ABC\). In each vertex of the figure, three sides meet, as in each vertex of the tetrahedron, three edges meet.

Similarly, using central projection, we can obtain from a regular octahedron a figure consisting of seven triangles, in each vertex of which four sides meet (Fig. 16), and from a regular icosahedron, a figure consisting of 19 triangles, in each vertex of which five sides meet (Fig. 17).

There is no figure that meets the conditions of the problem and differs from those shown in Figs. 15-17, as it would correspond to a regular polyhedron different from the three mentioned above, and such does not exist—compare with what is said in topics 70-80 (pp. 80-91) of the book "Mathematical Kaleidoscope"*) about so-called Platonic solids (or "regular polyhedra"): regular polyhedra with triangular faces exist only in three types, corresponding to Figs. 14-16; the corresponding values of the number \(F\) of parts of the plane (= the number of faces of the polyhedron) are 4, 8, and \(20^{**}\).