The Theorems of Ceva and Menelaus

Menelaus's Theorem
If L is a line not through any vertex of a triangle ABC such that
L meets BC in P, AC in Q, and AB in R,
then (AR/RB)(BP/PC)(CQ/QA) = -1.

Ceva's Theorem
If X is a point not on any side of a triangle ABC such that
AX and BC meet in P, BX and AC in Q, and CX and AB in R,
then (AR/RB)(BP/PC)(CQ/QA) = 1.

Note that these theorems involve the signed ratios of three
distinct points on a line. One way to define these is as follows:
If A, B and X are points on a line L, then
(AX/XB) =  |AX|/|XB| if X is between A and B, and
(AX/XB) = -|AX|/|XB| otherwise.

Although we could make a similar definition in hyperbolic geometry, the quantity
is not of much interest. After the section on the trigonometry of h-triangles,
we ought to consider using a hyperbolic function of the hyperbolic lengths.
The correct choice turns out to be sinh.

Definition
If A, B and X are distinct points on an h-line, then their hyperbolic ratio is
h(A,X,B) =  sinh(d(A,X))/sinh(d(X,B)), if X is between A and B, and
h(A,X,B) = -sinh(d(A,X))/sinh(d(X,B)), otherwise.

Basic Properties of Hyperbolic Ratio

(1) h(A,X,B) = -h(B,X,A),
(2) if X is between A and B, then h(A,X,B) ε (0,1),
(3) if X is on AB, beyond B, then h(A,X,B) ε (-∞,-1),
(4) if X is on AB, beyond A, then h(A,X,B) ε (-1,0).

Note that the value of h(A,X,B) determines the position of X relative to A and B.

Proof
(1) is an obvious consequence of the definition of h.
(2),(3) and (4) all follow from property D3, and the fact that
the function sinh is increasing.

With this definition, the appropriate Sine Rule leads to

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 BC in Q, AC in R, and AB in P,
then h(A,P,B)h(B,Q,C)h(C,R,A) = -1.

Proof of Menelaus's theorem

Ceva's Theorem for Hyperbolic Triangles
If X is a point not on any side of an h-triangle 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.

As in the euclidean case, Ceva's Theorem is a consequence of that of Menelaus.

Proof of Ceva's theorem

In many ways, the converses are more useful. These depend on the fact that,
for given points A and B, a point X on the h-line AB is determined uniquely by
the hyperbolic ratio h(A,X,B).

The Hyperbolic Ratio Theorem
If Xand Y are points on the h-line AB such that h(A,X,B) = h(A,Y,B), then X = Y.

Proof of the hyperbolic ratio theorem

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.

Proof of the converse of Menelaus's theorem

The converse of Ceva's Theorem needs a little more care. This is so because,
in hyperbolic geometry, two h-lines may be intersecting, parallel or ultraparallel.

Converse of Ceva'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,
and two of the h-lines CP, BR and AQ meet,
then all three are concurrent.

Proof of the converse of Ceva's theorem