Under which conditions there is an intersection point for all bisectors of a pyramid?
A bisector of a pyramid initiating from a vertex is a line which has the same angle with all edges who are neighbor of that vertex.(A natural generalization of bisectors in triangle).
These lines meet at the same point iff there is a sphere tangent to all six edges of the tetrahedron (for each vertex, such line clearly contains all possible centres of spheres tangent to the three edges adjacent to that vertex), which is not always the case. Tetrahedra with this property has several equivalent descriptions, see Theorem 1 in http://forumgeom.fau.edu/FG2007volume7/FG200703.pdf .