Inversion in Projective Geometry

Question

1. Let C1 and C2 be two circles both passing through the point S. Let C1 be inside C2. Prove that you can construct a circle of inversion i such that I(C2) = C1.

Answer 1

Let j be the inversion about a circle C₃ whose center is the point where C₁ and C₂ touch each other. Then j(C₁) and j(C₂) will be straight lines L₁ and L₂ that a parallel to each other. Let L₃ be the straight line half way between those two. Let k be inversion about L₃. Then k(L₁) = L₂ and k(L₂) = L₁. Now let C₃ = j(L₃). The C₃ is is the circle you're looking for. Inverting C₁ about C₃ yields C₂. The reason for that should become clear if you go through these steps carefully.