the arcs theorem

 the arcs theorem Suppose that K is a horocycle or hypercircle, and that A,B are points on K. Then, for any C on K other than A and B, (1) E2(A,B,C) is independent of C, and (2) E(A,B,C) < 0 if and only if C lies on the finite arc AB. proof (1) We have 1 - E2(A,B,C) = J(A,B,C)s2(AB). For A,B,C lie on K, J(A,B,C) = j(K), i.e. is fixed. It follows that, if we fix A and B, then E2(A,B,C) is constant. Since we have a horocycle or hypercycle, j(K) ≤ 0, so E2(A,B,C) ≥ 1. Indeed, if K is a horocycle, then j(K) = 0, so E2(A,B,C)=1. (2) If we vary C along a particular arc, then J(A,B,C) is continuous. But its values are the E(A,B,C) where E2(A,B,C) is fixed, and non-zero. It follows that E(A,B,C) is constant along each arc. For an infinite arc, we can choose C so that d(A,C) is larger than d(A,B). Then s(AC) > s(AB). It follows that E(A,B,C) < 0 for this C, and hence for all C on the arc. For a finite arc, we use the symmetry lemma to pick the point C where the hyperbolic bisector of AB cuts K. Since we have a horocycle or a hypercycle, we have H(s(AB),s(BC),s(CA)) ≤ 0. It follows that one of the arguments is at least as large as the sum of the others. For this C, s(AC) = s(BC), so we must have s(AB) ≥ s(AC)+s(BC). Then we have s2(AB) ≥ (s(AC)+s(BC))2 > s2(AC)+s2(BC), so E(A,B,C) < 0. By the continuity argument, this holds for all C on the finite arc AB. Note We have not established the continuity of J(A,B,C) with respect to C. The formula is quite complicated, and certainly has poles when C = A or B. Rather than get involved with these matters, we can give a proof which is purely geometric. Alternative proof of (2) Let H be the hyperbolic circle on AB as diameter. Then H cuts K at A and B, and hence, as i-lines, nowhere else. Thus, H divides K into its three arcs. As H cannot contain points arbitrarily close to the boundary C, the finite arc of K must be that lying inside H. Thus, if C is on th e finite arc, C is inside H. By the hyperbolic semi-circle theorem, s2(AC)+s2(BC) < s2(AB). Then E(A,B,C) < 0, as required. If C lies on an infinite arc defined by A,B, then either A lies on the finite arc BC, so E(B,C,A) < 0, or B lies on the finite arc AC, so E(A,C,B) < 0. By the definition, at most one of E(A,B,C), E(B,C,A), E(A,C,B) < 0. It follows that, for C on an infinite arc, E(A,B,C) > 0. Since E2(A,B,C) is constant, it follows that E(A,B,C) is constant on each arc.