Tessellations

Three examples of particular tilings of the plane, namely those where - NOT all corners of the tile are vertices of the tiling. You might like to produce an elegant program to replicate one or all of these.

Can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

Go on a vector walk and determine which points on the walk are closest to the origin.