Smallest quadrilateral containing a set of points

I am not sure how to find the least area/ least perimeter triangles that contain a set of points on the plane. The guess for both would be that the triangle has an edge coincident with at least one side of the convex hull of points so a plane sweep kind of approach would work.

An apparently more difficult question: How does one find the least area quadrilateral containing a set of points? The quadrilateral is allowed to be non-convex. If the quad is required to be convex, we might be able to manage with a plane sweep type of algo.

Non-congruent Tilings - 22

Some more tiling questions occured recently. Simply recording them. Hopefully they don't feature in earlier episodes of this series, the most recent being this and this.
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1. Can the plane be tiled by mutually non-congruent triangles all of equal area and with all three edges unique? If possible, one could demand the edge lengths to have an upper bound.
(This question was raised at this earlier episode) and here is another discussion that appears closely related but for the equal area requirement.

2. Can the plane be tiled by mutually non-congruent triangles all of same area and having one angle common?

3. Can the plane be tiled by mutually non-congruent triangles all having two sides common? No equal area constraint here.

4. Can the plane be tiled by mutually non-congruent triangles all with one side and one angle common? No equal area constraint.

And so on...there seems to be no end to possibilities...