Apr18-12, 09:29 AM
1. The problem statement, all variables and given/known data
Use the Poincare (disk) model to show that in the hyperbolic plane, there exists two points A, B lying on the same side S of a line l such that no circle through A and B lies entirely within S.
2. Relevant equations
The hint was to use this proposition:
A P-circle is a Euclidean circle in the disk, and conversly, but the P-center differes from the Euclidean center except when the center is the origin.
Basicallly in the disk circles still look like they normally would but the center might be different from what it would be in a eudclidean sense.
3. The attempt at a solution
I'm quite lost as to how to do this actually, I realize you should be able to do it by contradiction and just show that any circle you draw would have to either intersect the line l or it would have to touch the boundary of the disk. But it seems like you should be able to draw a circle quite easily that would go through two points, be inside another circle and not cross a line.
The book is eucliden and non-euclidean geometries development and history by marvin jay greenberg. Problem p-20.
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