Any point outside a circle may be specified, or named, by reference to the two tangent lines drawn from the point to the circle. Could this be the basis for creating a coordinate plane, albeit a cumbersome one, that's an alternative to the Cartesian plane?
For example, a point might be "named" by the measure of the arc subtended by the two tangents to the circle drawn from the point. The point m<30, subtended to the circumference of the circle at 320 and 350 degrees. Another point would be m<30, subtended to the diameter of the same circle at 20 and 50 degrees. Every point could be named, as in the Cartesian plane. m<45, m<60. etc would be easy beginning points. I've drawn a circle inscribing a 24-sided polygon as a starter, and used the vertices as tangent points for points outside the circle.
I haven't pursued the idea much further than this. Curious as to what anyone else thinks.