Suppose that ABC and PQR are h-triangles.
Then there is a hyperbolic transformation t which maps
A to P,
B to B' on the h-line PQ, on the same side of P as Q, and
C to C' on the same side of the h-line as R.
Note that, as t preserves angle and hyperbolic distance,
If h-triangles ABC and PQR have
If any pair of sides have equal hyperbolic length, then the
h-triangles are h-congruent by the (ASA) condition.
Assume that no two are equal (we will obtain a contradiction).
Let t be the transformation implied by the Basic Strategy.
By our assumptions, C' lies on PR, but C' ≠ R,
We have one of the two situations shown on the right.