Plane Geometry - Bruce E. Shapiro

204 SECTION 39. CIRCLESFigure 39.3: The tangent line theorem.By congruency, corresponding sides are congruent, hence OP = OR. Thismeans that R lies on Γ by definition of a circle.Since P ≠ R, this contradicts the fact that P is the only point that lies onboth l and Γ (since l is tangent to Γ at P ). Hence the RAA hypothesis iswrong and P = Q.Since OQ ⊥ l, then OP ⊥ l.(Proof of ⇐) Assume that l intersects Γ at P and that OP ⊥ l.Choose any point Q ∈ l that is distinct from P .Since OP ⊥ l at P , the shortest segment from O to l is OP . HenceOQ > OPBy definition of outside, Q is outside of Γ. Since Q was chosen arbitrarily,every point on l that is distinct from P is outside of Γ.Hence Γ ∩ l contains precisely one point P , which means l is tangent to Γat P .Corollary 39.8 Let Γ be a circle and l a line tangent to Γ at P . Thenevery point on l that is distinct from P is outside of l.Proof. This was proven by the next to the last line of the proof of theprevious theorem.Theorem 39.9 (Secant Line Theorem) Let l be a secant line that intersectscircle Γ = C(O, r) at distinct points P and Q. Then O lies on theperpendicular bisector of chord P Q.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012