What we have below is either an Ellipse or Hyperbola in the Poincaré disk. The foci are points A and C and the difference (or sum) of the focal radii are equal to the distance from points A to B. Points A, B, and C can be moved and when moved will result in a different Ellipse or Hyperbola.
* A is the center of the circle through point B
* If C is inside the circle we end up with an ellipse, else we end up with a hyperbola.
* Please give the construction a few seconds to plot the conic after moving the points around
* Point D is on the circle and can be moved around it if you like.
Note that the plot is in pink.