Proof of the hyperbolic circle theorem and corollary

The Hyperbolic Circle Theorem
If γεD, 0 < r < 1, and tεH(2), then t(K(γ,r)) = K(t(γ),r).

By the Origin Lemma, there is an h-inversion i mapping γ to 0 (and hence 0 to γ).
By the Lemma and the following remarks, i(Cr) = K(γ,r), so that

t(K(γ,r))   = t(i(Cr))
= toi(Cr)
= K(toi(0),r)   by the Lemma again
= K(t(γ),r)   as t(0) = γ

For 0 < r < 1, tεH(2) maps
the circle Cr = {z : |z| =r}
to the locus
{z : |z - t(0)|/|t(0)*z - 1| = r}.

This locus is K(t(0),r).

If z, w ε D and tεH(2), then D(t(z),t(w)) = D(z,w).

Suppose that D(z,w) = r. Then z ε K(w,r).
Applying t, t(z) ε t(K(w,r))
By the Theorem, t(K(w,r)) = K(t(w),r), so t(z) ε K(t(w),r).
Thus D(t(z),t(w)) = r = D(z,w), as required.

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