Proof of the Medians Theorem for Hyperbolic Triangles

An h-median of a hyperbolic triangle is an h-segment
joining a vertex to the h-midpoint of the opposite side.

The Medians Theorem for Hyperbolic Triangles
The h-medians of a hyperbolic triangle are concurrent.

Let ABC be an h-triangle, and let the h-medians be AQ, BR and CP.

Since P is the h-midpoint of BC,
h(A,P,B) = sinh(d(A,P))/sinh(d(P,B)) = 1,
and similarly h(B,Q,C) = h(C,R,A) = 1.
Hence h(A,P,B)h(B,Q,C)h(C,R,A) = 1.

The h-medians all lie within the h-triangle, so any two must meet.
Thus, by the Converse of Ceva's Theorem,
the h-medians AQ, BR and CP are collinear.

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