Let F be any set of five points in R3 so situated that no four of the points are coplanar, and that the line xy through any two x and y of the points has a unique intersection point xy* with the plane determined by the other three. Let F^ denote the family of all such xy*. Let S(F) denote the set of all X β F^ which are maximal with respect to the property that X is a subset of a plane in R3. For k > 2 an integer, let S(k; F) denote the family of all k-membered elements in S(F).
A family π of sets is said to be uniformly deep of depth d if and only if for every x β βͺ π there are exactly d distinct π β π for which x β π.
We establish the following result, and extend our ideas to general Euclidean spaces.