Affine geometry

The affine transformations of the upper half-plane include
# shifts , , and
# dilations ,

Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes to .
:Proof: First shift the center of to Then take and dilate. Then shift to the center of

Inversive geometry

Definition: .

can be recognized

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