Plane Geometry - Bruce E. Shapiro

SECTION 39. CIRCLES 209Since D(a)D(x) → 0 as x → a, we getlim f(x) = f(α)x→αTheorem 39.17 (Strong Circular Continuity) Let Γ = C(O, r) andΓ ′ = C(O ′ , r ′ ) be circles, let A be a point on Γ that is outside Γ ′ , and let Bbe a point on Γ that is inside Γ ′ . Then Γ ∩ Γ ′ contains precisely two points.Proof. Assume that A and B are not antipodal. If they are, by the continuityof f described in the previous theorem we can replace A with anotherpoint A ′ close to A ′ so that A is still outside Γ ′ .Chose a point C ∈ Γ such that A, B are on the same side of ←→ OC. Use thispoint to define a function f(x) like the one in theorem 39.16:f(x) : [0, µ(∠BOC)) ↦→ [0, ∞)byf(α) = O ′ Q(α)where Q(α) is the point on Γ such that α = µ(∠COQ(α)) (see figure 39.8).There exist numbers a, b ∈ (0, 180) such that Q(a) = A and Q(b) = B. ByFigure 39.8: Strong circular continuity.the previous theorem, f is continuous, with f(a) > r ′ and f(b) < r ′ .Hence by the intermediate value theorem there is a number x such thatf(x) = r ′ .Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.