Plane Geometry - Bruce E. Shapiro

Section 39CirclesDefinition 39.1 The circle C(O, r) with center O and radius r > 0 isthe set of all points P that are a distance r from O,C(O, r) = {P |OP = r}Definition 39.2 A chord of Γ = C(O, r) is a segment P Q joining twopoints P and Q on Γ. A line l that contains a chord of Γ is called a secantline of Γ.Definition 39.3 Points P and Q on Γ = C(O, r) are called antipodal ifP ∗ O ∗ Q, in which case the chord P Q is called a diameter.We will also use the term diameter to mean the length of the diameter; itshould be clear from the context whether we mean P Q or the length of P Q.Definition 39.4 Let Γ = C(O, r). Then point A is said to be outside ofΓ if OA > r, and inside of Γ if OA < r.Definition 39.5 A line l is tangent to a circle Γ = C(O, r) if Γ∩l containsprecisely one point. A segment AB ∈ l is tangent to Γ if l is tangent to Γand the point of tangency lies in the interior of AB.The following theorem tells us that given any line and any circle, the lineeither is a secant line, a tangent line, or it does not intersect the circle.Theorem 39.6 Let Γ = C(O, r) be a circle and l be a line.number of points in Γ ∩ l is either 0, 1, or 2.Then theProof. Suppose there are three distinct points A, B, C that all lie in Γ ∩ l(figure 39.2).201