Basic stategy Suppose that ABC and PQR are htriangles. Then there is a hyperbolic transformation t which maps A to P, B to B' on the hline PQ, on the same side of P as Q, and C to C' on the same side of the hline as R.
Note that, as t preserves angle and hyperbolic distance,



(SSS) condition If htriangles ABC and PQR have


Proof Let t be the transformation implied by the Basic Strategy. Then B' lies on the hline PQ. By the Basic Strategy, d(A,B) = d(P,B'). By (1), d(A,B) = d(P,Q), so d(P,B') = d(P,Q), and hence B' = Q. By the Basic Strategy and (2), (3)
(4) d(P,C') = d(P,R), and
Let Kp be the hcircle, with hcentre P, through R, and
By (4) and (5), C' lies on Kp and Kq Thus t maps ABC to PQR, i.e. the htriangles are hcongruent.
