Before revision:  As for finding smallest-area triangles, we show that finding the largest smallest-area triangle is NP-hard , but that the smallest possible area triangle for a set of arbitrary length parallel line segments can be found in O(n^2) time.
After revision:  As for finding smallest-area triangles, we show that finding the largest smallest-area triangle is NP-hard ; but minimizing the smallest triangle can be done in O(n^2) time.