Earlier we proved the following euclidean theorem
Van Obel's Theorem
If the point X does not lie on any side of the euclidean ΔABC, and
AB meets CX in P, BC meets AX in Q and CA meets BX in R,
then
(AX/XQ) = (AP/PB) + (AR/RC),
(BX/XR) = (BP/PA) + (BQ/QC),
(CX/XP) = (CQ/QB) + (CR/RA).


There is a hyperbolic analogue, proved in a similar fashion, but now
each ratio depends on more than two of the ratios in which the cevains
cut the sides. Of course, we use the hyperbolic ratios.
van Obel's Theorem in hyperbolic geometry
If the point X does not lie on any side of the hyperbolic ΔABC, and
AB meets CX in P, BC meets AX in Q and CA meets BX in R,
then
h(A,X,Q) = cosh(d(B,Q)h(A,P,B)+cosh(d(Q,C))h(A,R,C),
h(B,X,R) = cosh(d(A,R))h(B,P,A)+cosh(d(R,C))h(B,Q,C),
h(C,X,P) = cosh(d(A,P))h(C,R,A)+cosh(d(P,B))h(C,R,B).
proof


In the euclidean case, we can deduce that the centroid cuts each median
in the same ratio, 2. The hyperbolic case is rather different.
Theorem
If AQ, BR, CP are the hyperbolic medians of the hyperbolic triangle ABC,
and X is the hyperbolic centroid (the concurrence of the medians), then
h(A,X,Q)=2cosh(½d(B,C)),
h(B,X,R)=2cosh(½d(C,A)),
h(C,X,P)=2cosh(½d(A,B)).
These follow at once from van Obel's Theorem. For the first, we have
h(A,P,B) = h(A,R,C) = 1, and d(B,Q) = d(Q,C) = ½d(B,C). The others
are similar.
Note that these hyperbolic ratios are, in general, different. Also, since
cosh(a) > 1, each ratio is greater than 2.

