Menelaus's Theorem for Hyperbolic Triangles If L is an hline not through any vertex of an htriangle ABC such that L meets AB in P, BC in Q, and CA in R, then h(A,P,B)h(B,Q,C)h(C,R,A) = 1.


Proof A little consideration shows that, if we change the labels of the vertices, then either the factors are simply permuted, or are inverted and permuted. Since our aim is to show that the product is 1, the labelling of the vertices of the htriangle is immaterial.
It follows that there are two cases, depending on the position of the hline
Observe that either one cut is external, or all three are external.
Similarly, from the htriangles BPQ and CRQ,
Observe that:
Some elementary algebra now yields
Case 2 Note. Essentially the same argument works in the euclidean case.


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