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.


Justification By the Interchange Theorem, there is an hinversion u mapping A to P. Suppose that u maps B to B_{1}, and C to C_{1}. Now let v be the hinversion in the hline V bisecting <B_{1}PQ. As P is on V, v fixes P. By the choice of V, v maps B_{1} to a point B' on PQ on the same side as Q. Suppose that v maps C_{1} to C_{2}. If C_{2} is on the same side of PQ as R, we take t = vou so C' = C_{2}. If C_{2} is on the opposite side of PQ from R, let w be hinversion in PQ. Then w fixes P and B' (as they lie on PQ). Suppose that w maps C_{2} to C'. This lies on the same side of PQ as R since it lies on the opposite side from C_{2}. In this case, we take t = wovou.


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