Polar coordinates: r = 1/(1+2costheta)

Working through this equation, I'm happy with identifying asymptotes and points at theta = 0, Pi/2, Pi, 3/2Pi.
So it crosses x at 1/3 and 1 and y at 1 and -1 with asymptotes theta =2/3 Pi and 4/3 Pi (where cos =-1/2)
Now when you plot it you get symmetry half way between x =1/3 and 1 and you get asymptotes that are parallel to those from the origin, but centred at the line of symmetry.

I have a disconnect as to how I can justify the asymptotes running through x=2/3 as opposed to being at zero, since all the maths works from the origin. But the positive r line goes from y=-1 to x=+1/3 to y=1 so crosses the asymptotes as defined from the origin.