Poncelet's theorem in hyperbolic geometry

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The axioms of projective geometry are also satisfied in hyperbolic geometry, with the exception that two lines do not necessarily intersect (not even on the boundary of the Poincare disk). Therefore, as long as we only consider lines that are guaranteed to intersect then theorems from projective geometry will also be valid in hyperbolic geometry.

An example is Poncelet's theorem. Below is an animation of Poncelet triangles in hyperbolic geometry. The inner circle is the hyperbolic incircle of the hyperbolic triangle ABC, and the outer circle is the hyperbolic circumcircle (note that the circumcircle does not always exist for every choice of A, B, C since the hyperbolic circumcentre may move outside the Poincare disk). You can move the points A, B, C to change the triangle. Poncelet's theorem says that for any position of the point P on the circumcircle, the triangle PQR will be a Poncelet triangle.

The animation below shows the hyperbolic Poncelet theorem for a seven-sided hyperbolic polygon. The original configuration describes a Poncelet polygon with one vertex at a point P on the outer circle. By moving the points O (hyperbolic centre of the inner circle) and Q (point on the inner circle) you can change the position of the inner circle, and by moving the points A and B you can change the position of the outer circle to see that the polygon either (a) closes up for all positions of P on the outer circle, or (b) does not close up for any position of P on the outer circle.