Plane Geometry - Bruce E. Shapiro

210 SECTION 39. CIRCLESLet D be the point on Γ corresponding to x. Since f(x) = r ′ we haveD ∈ Γ ′ , hence Γ and Γ ′ intersect at D.This proves that there is at least one point in Γ ∩ Γ ′ .Figure 39.9: If Γ and Γ ′ share a common tangent line then they are eitheron the same side of the line (top) or the opposite sides (bottom).Suppose that D is the only point in Γ ∩ Γ ′ (RAA hypothesis).Then the two circles share a common tangent line l (figure 39.9).Either they are on the same side of l or on different sides of l.If the two circles are on opposite sides of the l then every point on Γ (exceptfor D) is outside of Γ ′ , contradicting the fact that B is inside Γ ′ .Hence the two circles cannot be on opposite sides of l. Then every pointof one circle (except for D) is inside the other circle. This contradicts thefact that A is outside Γ ′ and B is inside Γ ′ .Since the two circles cannot either be on the same side or the oppositesides of the tangent line, the tangent line cannot exist. This means we« CC BY-NC-ND 3.0. Revised: 18 Nov 2012