VMO 1996 Problem 2

SABC is a tetrahedron. DAB, EBC, FCA are triangles in the plane of ABC congruent to SAB, SBC, SCA respectively. O is the circumcenter of DEF. Let K be the exsphere of SABC opposite O (which touches the planes SAB, SBC, SCA, ABC, lies on the opposite side of ABC to S, but on the same side of SAB as C, the same side of SBC as A, and the same side of SCA as B). Show that K touches the plane ABC at O.