<Solid-2025-266>, hint

Let H be the foot of the  altitude , the intersection of diagonal AC and BD.

Since SA, SC, SH lie on the same plane, AA' cut PH at  Q and AA'ㅗSC,

Thus A' lie on the sphere S" of diameter SH.

Suppose CC' cut PH at Q', then Q' lie on the sphere of diameter SQ', different from S",

which contradict the proposition A',C' lie on the same sphere.

Hence Q=Q'. Finally S,A',B',C',D' lie on the same sphere and AA', BB', CC', DD' concur

at Q, same point.