van Obel's's Theorem in euclidean geometry

 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). 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.