<Solid-2023-237>,hint

Let T'C be point where S' touches the plane DAB, T'B' T'A similarly defined.

T"C, T"B, T"A be point where S" touches the plane DAB, DCA, DCA .

Let O',O" be the circumcenters of triangles T'AT'BT'C,  T"AT"BT"C.

Since the above two triangles are similar,  DO' /DO"= O'T'A/O"T"A. and K,L,T are perpendicular projections of ,O',O",D unto the plane ABC. Points K,L,T are collinear.