With recent interest in the Hadwiger-Nelson problem on the chromatic number of the plane, thanks to de Grey's theorem that $CNP\ge5$, I've been looking at unit-distance embeddings of various graphs with chromatic number $\chi\ge 4$.
For example, looking at the unit-distance embedding of the Moser spindle, it's clear that three of the vertices form an isosceles triangle with sides $(1,2\sqrt{3},2\sqrt{3})$.
However, looking at the unit-distance embedding of the Golomb graph, I can't see a quick way to deduce the coordinates of the inner triangle.
All of the edges in the graph above are of length $1$.
Is there a quick "Elements-style" explanation of the coordinates of the vertices in the unit-embedding of Golomb graph? Is there a straightforward compass-and-straightedge construction?
Start by constructing the middle equilateral triangle and the blue point in its center.
Now you can put down the outer green point, because it has a known distance from the blue and inner yellow points.
Finally complete the outer hexagon.
(If you want coordinates in a system where the outer hexagon to have an axis-parallel side, you now need to rotate the entire drawing appropriately).