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Good thinking, Janaandbillysc . Perhaps what you are interested in is polar coordinates.

Thank you so much.  This very helpful.  A vector together with a distance and the intersection of two tangents are two ways of naming the same point.  Perhaps they are both polar coordinates.  I'll keep thinking about their similarities and differences.

This is an edited version of my post above that more clearly expresses what I was trying to get at:

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 "named" (270,30) would the is the intersection of the tangent lines at 270 degrees and 30 degrees of the 360 degree circle. Another point, (20,50),would be the intersection of the tangent lines at 20 degrees and 50 degrees. Every point could be named, as in the Cartesian plane. 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've drawn a triangle with vertices at (270, 30), (300,60), and (300,30). It's a 45, 45, 90 degree right triangle.

I think I understand the method you are proposing. You would have to indicate the radius of the circle as well, along with the two angles. And there should be a method to cover the points inside the circle too, so as to ensure all points in 2-D space can be specified using your method.

In any case it's an interesting thought.

Yes, to your suggestions.  I think we're on the same page.

As to specifying the radius of the circle, mightn't it be a "unit" circle, in the same way that a square in the Cartesian plane is a 'unit' without being more particularly specified?

As to points inside the circle, I'm not so sure.  As I make the circle smaller and smaller, the limit of the radius approaches zero, and the circle becomes a point, but short of that even a very, very small circle will have points inside, theoretically I suppose, as many points as any bigger circle, that will need to be specified.

Consider the point (300.60) which names the intersection of two tangents of a circle drawn at 300 degrees and 60 degrees of the circle.  Now, extend these tangent lines in both directions.  Any (every) circle, from the smallest to the largest, drawn "below" the intersection, that has these two lines as tangents will have the same point in common named (300,60).  Those circles, from the smallest to the largest, drawn 'above' the intersection that have these two lines as tangents will have the very same point as above in common named (120, 240).

This may speak to not needing to specify names for points inside the circle.  There may be a place for an integral here.  As the size of the circle approaches the limit of a point, it will still have the point (300,60) in common with all larger circles.

Again, I'm not 'fluent' in calculus yet, but I think there might be something here.  Any thoughts?

To clarify:  For circles 'below' the intersection, the point of intersection is named (300,60); for circles 'above' the  intersection the point of intersection is named (120,240).  Two names for the same point.  As the size of the circle ('below' the point) approaches the limit of a point, it will still have the point (300,60) in common with all larger circles 'below' the point.

One pair of intersecting tangent lines (one point, e g 300,60) can be common to many circles.  Two pair of intersecting tangent lines (two points, e g 300,60, and 270,30) specify one, and only one, circle. (I think).