Plane Geometry - Bruce E. Shapiro

224 SECTION 41. EUCLIDEAN CIRCLESFigure 41.5: Proof of the Central Angle Theorem when the center of thecircle is in the interior of the inscribed angle.Definition 41.12 Let B be a circle and let O be a point that does not lieon B. The power of O with respect to B is defined as follows: Choose anyline l through O that intersects B. If l is a secant line that intersects B attwo points P, Q, then the power is the product (OP )(OQ). If l is tangentto B at P then the power of O is defined as (OP ) 2 .Theorem 41.13 The power of a point is well defined, i.e, the same valueis obtained regardless of which line is used, so long as at least one point onthe line intersects the circle. aa Euclid book 3, Proposition 36.Proof. (Case 1) Suppose that O is center of Γ(A, r). Then any line throughO intersects Γ at two points P and Q such that OP = OQ = r. Hence(OP )(OQ) = r 2 regardless of which line is chosen through O.(Case 2) O is inside Γ and is not the center of Γ = C(A, r).Let l be any line through O. Then it intersects Γ in two poins Q and R.Let S and T be the points where line ←→ AO intersects Γ.Then α = β since both inscribed angles intercept the same arc (correspondingto central angle γ).Similarly, η = θ because both inscribed angles intercept the same arc (cor-« CC BY-NC-ND 3.0. Revised: 18 Nov 2012