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(Edit:) I think I may have misunderstood the question. Here are some results for polygons created by finite samples from an infinite family of lines:

Let each line have a point on the unit circle, at angle θ anticlockwise from the x-axis, where 0 ≤ θ < 1. Each line has another point also on the unit circle, at angle φ anticlockwise from its start angle. (Here, φ is the golden ratio, (1+√5)/2.) We get this:


This image shows lines (in blue) where the angles θ were generated by the start of the quasirandom sequence (nφ mod 1). This countable infinite sequence is guaranteed not to have any parallel lines. In this family, different angles also guarantee non-coincident intersections.

The image is marked up with the largest possible triangle (in red), with area ≈0.017. So, this is a case with an upper bound on area. In fact, all intersections (and therefore all polygons) fall inside that triangle plus the curved parts just below it.

From a random sample of 100,000 examples, this is the distribution of triangle areas:  

I agree that it seems unlikely that areas could all be both finite and nonzero, but I have no proof.

(Edit:) I think I may have misunderstood the question. Here are some results for polygons created by finite samples from an infinite family of lines:

Let each line have a point on the unit circle, at angle θ anticlockwise from the x-axis, where 0 ≤ θ < 1. Each line has another point also on the unit circle, at angle φ anticlockwise from its start angle. (Here, φ is the golden ratio, (1+√5)/2.) We get this:


This image shows lines (in blue) where the angles θ were generated by the start of the quasirandom sequence (nφ mod 1). This countable infinite sequence is guaranteed not to have any parallel lines. In this family, different angles also guarantee non-coincident intersections.

The image is marked up with the largest possible triangle (in red), with area ≈0.017. So, this is a case with an upper bound on area. In fact, all intersections (and therefore all polygons) fall inside that triangle plus the curved parts just below it.

From a random sample of 100,000 examples, this is the distribution of triangle areas:  

I agree that it seems unlikely that areas could all be both finite and nonzero, but I have no proof.

Here are some results for a simple family of lines:

Let each line have a point on the unit circle, at angle θ anticlockwise from the x-axis, where 0 ≤ θ < 1. Each line has another point also on the unit circle, at angle φ anticlockwise from its start angle. (Here, φ is the golden ratio, (1+√5)/2.) We get this:


This image shows lines (in blue) where the angles θ were generated by the start of the quasirandom sequence (nφ mod 1). This countable sequence is guaranteed not to have any parallel lines. In this family, different angles also guarantee non-coincident intersections.

The image is marked up with the largest possible triangle (in red), with area ≈0.017. So, this is a case with an upper bound on area. In fact, all intersections (and therefore all polygons) fall inside that triangle plus the curved parts just below it.

From a random sample of 100,000 examples, this is the distribution of areas:  

I agree that it seems unlikely that areas could all be both finite and nonzero, but I have no proof.

(Edit:) I think I may have misunderstood the question. Here are some results for polygons created by finite samples from an infinite family of lines:

Let each line have a point on the unit circle, at angle θ anticlockwise from the x-axis, where 0 ≤ θ < 1. Each line has another point also on the unit circle, at angle φ anticlockwise from its start angle. (Here, φ is the golden ratio, (1+√5)/2.) We get this:


This image shows lines (in blue) where the angles θ were generated by the start of the quasirandom sequence (nφ mod 1). This countable infinite sequence is guaranteed not to have any parallel lines. In this family, different angles also guarantee non-coincident intersections.

The image is marked up with the largest possible triangle (in red), with area ≈0.017. So, this is a case with an upper bound on area. In fact, all intersections (and therefore all polygons) fall inside that triangle plus the curved parts just below it.

From a random sample of 100,000 examples, this is the distribution of triangle areas:  

I agree that it seems unlikely that areas could all be both finite and nonzero, but I have no proof.

Here are some results for a simple family of lines:

Let each lines have a point on the unit circle, at angle θ anticlockwise from the x-axis, where 0 ≤ θ < 1. Each line has another point also on the unit circle, at angle φ anticlockwise from its start angle. (Here, φ is the golden ratio, (1+√5)/2.) We get this:


This image shows lines (in blue) where the angles θ were generated by the start of the quasirandom sequence (n φ) mod 1. This countable sequence is guaranteed not to have any parallel lines. In this family, different angles also guarantee non-coincident intersections.

That image is marked up with the largest possible triangle (in red), with area ≈0.017. So, this is a case with an upper bound on area. In fact, all intersections (and therefore all polygons) fall inside that triangle plus the curved parts just below it.

From a random sample of 100,000 examples, this is the distribution of areas:  

I agree that it seems unlikely that areas could all be finite and nonzero. In fact, is it true that if there is a finite upper bound on area, then all lines pass through some finite circle and the most common area is asymptotically zero?

Here are some results for a simple family of lines:

Let each line have a point on the unit circle, at angle θ anticlockwise from the x-axis, where 0 ≤ θ < 1. Each line has another point also on the unit circle, at angle φ anticlockwise from its start angle. (Here, φ is the golden ratio, (1+√5)/2.) We get this:


This image shows lines (in blue) where the angles θ were generated by the start of the quasirandom sequence (nφ mod 1). This countable sequence is guaranteed not to have any parallel lines. In this family, different angles also guarantee non-coincident intersections.

The image is marked up with the largest possible triangle (in red), with area ≈0.017. So, this is a case with an upper bound on area. In fact, all intersections (and therefore all polygons) fall inside that triangle plus the curved parts just below it.

From a random sample of 100,000 examples, this is the distribution of areas:  

I agree that it seems unlikely that areas could all be both finite and nonzero, but I have no proof.