## Task 33/71

Under what conditions does a sphere exist for a tetrahedron that touches all the edges of the tetrahedron (excluding extensions)?
The four vertices of the tetrahedron are $P_{1}, P_{2}, P_{3}$, and $P_{4}$, the length of the edge $P_{i} P_{j}$ (with $i, j=1 ; 2 ; 3 ; 4$) is $a_{i j}$. The sought sphere intersects (if it exists) the side faces of the tetrahedron in the incircles of these triangles. The incircles touch each other in pairs at a point on the edge $P_{i} P_{j}$, denoted by $B_{i j}$. The length of the segment $P_{1} B_{12}$ is calculated

1. in the triangle $P_{1} P_{2} P_{3}$ as $P_{1} B_{12}=\frac{a_{12}+a_{13}+a_{23}}{2}-a_{23}$
2. in the triangle $P_{1} P_{2} P_{4}$ as $P_{1} B_{12}=\frac{a_{12}+a_{14}+a_{24}}{2}-a_{24}$

Equating the right sides yields, after some transformations, $a_{13}+a_{24}=a_{14}+a_{23}$. Similarly, by calculating the segment $P_{1} B_{13}$, we obtain $a_{12}+a_{34}=a_{14}+a_{23}$.

Combining both equations, we obtain the necessary condition for the existence of a sphere that touches all edges of the tetrahedron:

$$
a_{12}+a_{34}=a_{13}+a_{24}=a_{14}+a_{23}
$$

The three sums of the lengths of any two non-intersecting edges must all be equal. It remains to be shown that the fulfillment of this condition is also sufficient. This, however, follows immediately from the fact that all the inferences and transformations performed are reversible.