I have seen that ellipse and hyperbola have a lot in common. One thing that is bugging me is the fact that I know a lot of the special case of ellipse where the major and minor axes are equal (circle) but I know next to nothing about special case of hyperbola where the major and minor axes are equal (rectangular hyperbola).
It would be really nice if someone could give me some food for thought.
In the case of a circle, not only the axes are equal, but that also forces its foci to coincide. In the case of a rectangular hyperbola that doesn't happen, hence you cannot expect to find as many "great geometric properties" as in the case of the circle.
For instance, the inscribed angles properties of a circle don't hold for a rectangular hyperbola.