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Theorem HA5
The point PεD is equidistant from the hyperbolic lines K and L if and only if
P lies on an axis for K,L.
proof
Let the hyperbolic perpendiculars from P meet K at U, L at V respectively.
Suppose that P lies on an axis H for K,L, and let h denote inversion in H.
As h maps K to L, U' = h(U) is on L. As h fixes P and preserves angles,
PU' is perpendicular to L, so U' = V. Thus d(U,P) = d(U',P) = d(V,P).
Now suppose that P is equidistant from K and L. Let M denote the hyperbolic
bisector of <UPV. As d(U,P) = d(V,P), inversion in M swaps U and V. Since
K,L are the unique perpendiculars to PU at U and PV at V respectively, this
inversion swaps K,L, so M is an axis.
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