It is tempting to assume that the solution touches two corners from the set of
all rectangle corners. This is not true. It is safe to assume that it touches
one. Guess this corner v.

Although it is not true that it will touch a second corner, it is true that it
will touch a side s such that v is not one of the endpoints of s. Guess s.

Let one endpoint of the solution slide along s, while ensuring the line to go
through v. Project the time when the line hits and leaves every rectangle onto
s and do a greedy sweep through in the end.

Running time O(n^3).