466 I. ASYMPTOTIC CONES AND SHARAFUTDINOV RETRACTION
by t, taking the limit as t---+ oo, and applying part (i), we obtain
( a^2 + b^2 - 2ab cos ( d 00 ('r1, 12)))
112:S ( a^2 + c^2 - 2accos ( d 00 ('r1, /3)))
112+ ( c^2 + b^2 - 2cbcos ( doo (13, 12)) )112
.We now show that(I.9)In the Euclidean plane, draw two adjacent triangles
L':>AOC and L':>COBwith side lengths L ( OA) =a, L ( OC) = c, L (OB) =band angles LAOC =
doo (11,/3) and LCOB = d 00 (13,/2). Then, by the law of cosines,
L(Ac) = (a^2 +c^2 -2accosdoo(/1,13))1
12andL (CB) = ( c^2 + b^2 - 2cb cos doo (13, 12) )112
.Choosing c so that the points A, C, and B lie on one line, we haveThus( a^2 + b^2 - 2abcos ( d 00 ('r1,12)))
112:SL(AC)+L(CB)
= L (AB)
= ( a^2 + b^2 - 2abcos ( d 00 ('r1, /3) + d 00 ('y3,12)))
112
.COS ( doo (11 , /2)) 2:: COS ( doo ('r1 , /3) + doo (13 , /2))and the triangle inequality (I.9) follows since d 00 (11,12) :::; 1r. DThe ideal boundary ( M ( oo) , d 00 ) of ( M, g) is defined to be the metric
space induced by (Ray M (p) , d 00 ) as given in ( G .3). In particular,
(I.10)and /1 rv /2 if and only if d 00 ('r1,12) = 0. Note that d 00 ('r1,12) = 0 if and
only iflim d(ti(t) ,/2 (t)) = O.
t---+oo t