TITLE: How to cram a segment and a triangle as tightly as possible?
QUESTION [6 upvotes]: You are given a triangle and a segment on a plane. The segment is longer than any of the triangle's sides. How can you translate and/or rotate the triangle or the segment such that the resulting convex hull has minimal area and the segment doesn't cut the triangle?
The segment has to be tangent to the triangle otherwise moving it towards the triangle until it collides with it will reduce the area of the convex hull.
I thought it would be pretty simple to prove that the segment would have to overlap one of the triangle's sides but failed so far. Could you help me to prove or disprove this?

REPLY [5 votes]: Your second paragraph shows that one side of the triangle must be part of the (moved) line segment.
If it touches at just a vertex you can rotate it to reduce the distance to both the other vertices of the triangle. That will reduce the area of the convex hull.
Then the minimum area of the convex hull will occur when  the longest side of the triangle lies on the segment, and it does not matter where on that segment the side lies.
To see why, note that the convex hull is a triangle with third vertex the other vertex of the triangle. These all have the same altitude, so the same area. The area will be smallest for the side with the smallest altitude. That's the longest side.
So the area of a minimal convex hull is half the product of the length of the segment and the shortest altitude of the triangle.