Throughout the solution we suppose that some points or lines can be infinite. Denote touch-points of sphere with $SA$, $SB$, $SC$, $SD$, $TA$, $TB$, $TC$, $TD$ by $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$, $A_{2}$, $B_{2}$, $C_{2}$, $D_{2}$ respectively. Points $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ lie in one plane $\alpha$ (it's a polar plane of $T$ wrt the sphere); analogously $A_{2}$, $B_{2}$, $C_{2}$, $D_{2}$ lie in one plane $\beta$. Let line $\ell$ be the intersection of these planes. Also let $AB\cap A_{1}B_{1}=E$, $AB\cap A_{2}B_{2}=E'$ (it's clear that both these points exist). By the Menelaus theorem $\frac{|AE|}{|EB|} =\frac{|AA_{1}|}{|B_{1}B|} =\frac{|AA_{2}|}{|B_{2}B|} =\frac{|AE'|}{|E'B|}$ and thus $E=E'$, i.e. $AB$ intersects $\ell$; analogously $AD$ and $BC$ intersect this line. Finally denote by $\omega$ plane through lines $\ell$ and $AB$. $C$ lie in one plane with $B$ and $\ell$, i.e. it lies in $\omega$. Analogously $D$ lies in $\omega$, so $\omega$ passes through all four points, and we are done.