Label the tetrahedron $ABCD.$ The edges $BC,CA,AB,DB,DC,DA$ are tangent to the sphere at $P,$ $Q,$ $R,$ $S,$ $T,$ $U,$ respectively. According to the topic Tangent points of a quadrilateral to a sphere are coplanar, we get that $P,T,U,R$ an $R,S,T,Q$ are coplanar $\Longrightarrow$ $PU \cap RT \ne \emptyset$ and $QS \cap RT \ne \emptyset.$ $PU,QS,RT$ are projected from $D$ on the plane $ABC$ into $AP,BQ,CR,$ but since $\odot(PQR)$ is the incircle of $\triangle ABC$ (it could also be an excircle) then $AP,BQ,CR$ are the Gergonne cevians of $\triangle ABC$ concurring at its Gergonne point $G.$ Therefore, intersections $PU \cap RT$ and $QS \cap RT$ lie on $DG$ $\Longrightarrow$ $PU,$ $QS,$ $RT$ concur.