A tesselation from the radii of three intersecting circles.
Remember to click "Show the Circles", or there won't be much to see for the construction.

This is often called a hexagon tesselation.
A few notes:
1) The constraints do not restrict the shaded regions to convex hexagons,
2) The shaded regions may overlap.
3) We can change the number of vertices enclosing a region, using the same points.
4) I say, we can use hexagons to determine two of the three points A, B, C. These hexagons are not the shaded regions in the final diagram.
...And Questions
Q1: What are the relationships among all three points A, B, C?
Q2: How many independent points are there? For example, suppose we fix point A at the origin, and can only move points B and C. Ryan says, 'hey, wait, I can still get every configuration, by dragging the workspace.' Can Ryan do better than this? For example, can he determine a coordinate space?
Q3: How can I keep the lines from intersecting?
Q4: ...and the shaded regions convex?