The Klein View of Geometry 
In euclidean, similarity and projective geometry, two points are sufficient to determine a line. In inversive geometry, there are infinitely many ilines through two given points. If A, B ε E, there is one extended line through A and B, but an infinite family of circles.
Here, we show that it needs three points of E^{+} to determine an iline. We shall
also The key is


The Fundamental Theorem of Inversive Geometry
If L = (α,β,γ) and L' = (α',β',γ') are lists of distinct points of E^{+}, Note that the element t lies in the subgroup I^{+}(2). There are several ways to prove this result. We offer a constructive proof, based on The (0,1,∞) Theorem
If L = (α,β,γ) is a list of distinct points of E^{+}, Proof of the fundamental theorem


The (0,1,∞) Theorem allows us to prove the promised results about ilines. We begin with a special case:
The points 0, 1 and ∞ lie on R, the extended real axis. Since any
iline through ∞


Theorem I3 (1) If α, β, γ are distinct points of E^{+}, then there is a unique iline through these points. (2) If L and M are ilines, then there is an inversive transformation t mapping L to M.
Proof If we ignore the point ∞ we get an unusual proof of a standard euclidean result:
Euclidean Theorem
Proof

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