What is the green line in the large diagram? I am pretty sure it does not go through the touch point of the small circle and the large one with the center at bottom left. And the pink line seems to be wrong too. Don't they all (green, pink and light blue) have to go through the small circle's center?

If you know that the lines must go through the small circle's center, it becomes a fairly simple geometrical problem with three Pythagorean equations and three variables (x, y, r).

The green line is the x=y diagonal. My guess is that the author tried to eyeball the diagram by moving the points G, H, I around on their respective circles, but when the point I was close to the diagonal, it got snapped to this position, which means that the circle through the three points isn't exactly tangent to the other three circles.

It's basically the equivalent of a rough sketch on paper, not an exact construction.