Before revision:  We show that for a given set of parallel line segments of equal length the largest possible area triangle can be found in O(n \log n) time , and for line segments of arbitrary length the problem can be solved in O(n ^2) time . Also, we show that the smallest largest-area triangle can be found in O(n^2 \log n) time .
After revision:  We show that for a given set of parallel line segments of equal length the largest possible area triangle can be found in O(n \log n) time , and for line segments of arbitrary length the problem can be solved in O(n \log n) time for unit segments); minimizing the largest triangle can be found in O(n^2 \log n) time .