Ivlev $\Phi$,

Given a tetrahedron in which a sphere can be inscribed, touching all its edges. Let the segments of the tangents from the vertices be $a, b, c$, and $d$. Is it always possible to form some triangle from these four segments? (Not all segments need to be used. It is allowed to form a side of the triangle from two segments.)
Let's take two circles $\beta$ and ү with radii 2 and 1, respectively, touching each other externally. We draw a common external tangent to both circles and construct a circle $\delta$ inscribed in the curvilinear triangle formed by both circles and the tangent. Clearly, the radius of this circle is less than 1, so a triangle cannot be formed from the radii of the three circles. Now, replace the line tangent to $\beta$ and $\gamma$ with a circle $\alpha$ with a radius greater than 4, touching them externally, and construct three spheres whose centers and radii coincide with the centers and radii of the circles $\alpha, \beta, \gamma$. Finally, construct a sphere with a radius equal to the radius of circle $\delta$, touching the other three. The centers of these spheres form a tetrahedron, and the points of their tangency lie on a sphere that touches all the edges. In this case, the segments $a, b, c, d$ are equal to the radii of the circles $\alpha, \beta, \gamma, \delta$, from which no triangle can be formed.

## Answer

Not always