Converse of Menelaus's Theorem for Hyperbolic Triangles
If P lies on the h-line 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 h-collinear.
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 h-triangle is immaterial.
The key is to show that an h-line 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 h-ray AB, then the h-line 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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