Converse of Menelaus's Theorem for Hyperbolic Triangles If P lies on the hline AB, Q on BC and R on CA such that h(A,P,B)h(B,Q,C)h(C,R,A) = 1, then P, Q and R are hcollinear.


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 hypothesis is that the product is 1, the labelling of the vertices of the htriangle is immaterial.
The key is to show that an hline through two of P,Q,R cuts the third side
Since the product is negative, at least one of the cuts is external.
If P lies on the hray AB, then the hline QP cuts AC, at R* say.
There are two other possibilities: In each case a similar application of Menelaus gives the result. Note. Essentially the same argument works in the euclidean case.


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