tangency lemma
Suppose that X is a point on C, and that K is a hyperbolic line not through X.
Then there is a unique horocycle J at X touching K.
Indeed, J touches K at the foot of the perpendicular from X to K.
proof
Probably the simplest way to see this is to invert the diagram in a circle with centre X.
Then C maps to an extended line L, and any horocycle at X to an extended line M'
parallel to L'. K maps to semi-circle K' with centre C on L'. It follows that there is just
one
such M' which touches K' (at the point where perpendicular through C cuts K').
Inverting again gives the result.
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