# The Theorems of Ceva and Menelaus

 Among the theorems on euclidean triangles, we have those 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. 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. 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. 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. 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.