Menelaus's Theorem for Hyperbolic Triangles
If L is an h-line not through any vertex of an h-triangle 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.
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 h-triangle is immaterial.
It follows that there are two cases, depending on the position of the h-line
Observe that either one cut is external, or all three are external.
Similarly, from the h-triangles BPQ and CRQ,
Some elementary algebra now yields
Note. Essentially the same argument works in the euclidean case.
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