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We show that a set A of lines in PG ( 4 , q ) , q even, is the set of secant lines of a parabolic (non-singular) quadric if and only if A satisfies the following three conditions: (I) every point of PG ( 4 , q ) lies on 0 , 1 2 q 3 or q 3 lines of A ; (II) every plane of PG ( 4 , q ) contains 0, 1 2 q ( q + 1 ) or q 2 lines of A ; and (III) every hyperplane of PG ( 4 , q ) contains 1 2 q 2 ( q 2 + 1 ) , 1 2 q 3 ( q + 1 ) or 1 2 q 2 ( q + 1 ) 2 lines of A .

Let E be a set of solids (hyperplanes) in PG ( 4 , q ) , q even, q > 2 , such that every point of PG ( 4 , q ) lies in either 0, 1 2 ( q 3 − q 2 ) or 1 2 q 3 solids of E , and every plane of PG ( 4 , q ) lies in either 0, 1 2 q or q solids of E . This article shows that E is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric Q ( 4 , q ) in an elliptic quadric.