A planar region C such that there is an 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 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.