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The geometry of similarity in the Euclidean plane or Euclidean space.
In Euclidean space, never.
But they can in non-Euclidean geometries.
I do not believe that such a quadrangle can exist in Euclidean space.
If they are coplanar in a Euclidean space, then yes. If they are not coplanar or not in Euclidean space, then not necessarily.
Euclidean space