Plane Geometry - Bruce E. Shapiro

SECTION 39. CIRCLES 203Figure 39.2: A circle cannot intersect a line at three distinct points, becauseotherwise all four of the indicated angles would have to be equal and smallerthan 90.Henceα + γ < 180But all three points A, B, C ∈ l, henceα + γ = 180This is a contradiction. Hence there cannot be three distinct points thatlie on both a line and a circle.Hence there cannot be more than 3 points that lie on both the line and thecircle; otherwise, if there were, pick any three of them. By the argumentgiven above, they cannot exist.Consequently the largest number of points that can lie in Γ ∩ l is two.Theorem 39.7 (Tangent Line Theorem) A line l is tangent to a circleΓ if and only if it is perpendicular to the diameter at the point of tangency.Proof. (⇒) Let Γ = C(O, r). Suppose that l is tangent to Γ at P . We needto show that l ⊥ OP .Drop a perpendicular line from O to l and call the foot Q. Suppose thatP ≠ Q (RAA Hypothesis).Then we can chose a point R ∈ l such that P ∗ Q ∗ R and P Q = QR..Triangles △OP Q and △ORQ share a common side (OQ); and since OQ ⊥ l(by hypothesis), ∠OQR = ∠OQP . Since R was constructed such thatP Q = QR, then two triangles are congruent by SAS.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.