Ceva's Theorem for Hyperbolic Triangles If X is a point not on any side of an htriangle ABC such that AX and BC meet in Q, BX and AC in R, and CX and AB in P, 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 essentially three cases, depending upon the position
Similarly, applying Menelaus's Theorem to the htriangle AQB, cut by CXP,
Dividing these two expressions and cancelling common factors, we get
Since, in this case, the sides are cut internally,
Cases 2 and 3 are almost identical, with just the signs of the h(.,.,.) to be
Note The proof is essentialy the same as for the euclidean case, with the


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