A planar region C such that there is a unique interior point that bisects all chords of C that passes through it may be termed *centrally symmetric*. It appears that such figures exist in non-Euclidean geometry as well. 

**Question:** Are these claims (shown to be valid in Euclidean plane at https://mathoverflow.net/questions/448712/a-claim-on-the-concurrency-of-area-bisectors-of-planar-convex-regions) meaningful and valid in the hyperbolic plane (assumption: area bisector and perimeter bisector are definitions that go over to hyperbolic geometry)?

- A planar convex region is centrally symmetric if and only if its area bisectors are all concurrent.

- A planar convex region is centrally symmetric if and only if its perimeter bisectors are all concurrent.