Plane Geometry - Bruce E. Shapiro

206 SECTION 39. CIRCLESFigure 39.5: Weak circular continuity (theorem 39.11): If A is inside of Γand B is outside of Γ, then ←→ AB must intersect Γ.Define f : [0, d] ↦→ [O, ∞) by f(x) = OD(x).By theorem 28.3 (Continuity of distance), f(x) is a continuous function.Since A is inside Γ, OA < r. HenceSince B is outside Γ, OB > r. Hencef(0) = OA < rf(d) = OB > rBy the intermediate value theorem there is some number u ∈ (0, d) suchthatf(u) = rSince the mapping between AB and [0, d] is one-to-one, there is some pointQ in AB such that OQ = r. Since f(u) = r, this means OQ = r, which inturn means Q is on Γ.Since 0 < r < d, then A ∗ Q ∗ B. Hence Q is in AB, and therefore ABintersects Γ.Corollary 39.12 If Γ is a circle and l is a line that contains a point A thatis inside Γ, then l is a secant line of Γ.Theorem 39.13 (Tangent Circles Theorem) If circles Γ = C(O 1 , r 1 )and Λ = C(O 2 , r 2 ) are tangent at P then points O 1 , O 2 , P are distinct andcollinear, and the two circles share a common tangent line at P .Proof. (Exercise.)« CC BY-NC-ND 3.0. Revised: 18 Nov 2012