Plane Geometry - Bruce E. Shapiro

SECTION 41. EUCLIDEAN CIRCLES 225Figure 41.6: Proof of the central angle theorem when the center of the circleis exterior to the inscribed angle.responding to central angle ζ).Also, ɛ = δ because they are vertical angles.Hence (AAA) △OQT ∼ △OSR.By the similar triangle theoremOROS = OTOQ(OR)(OQ) = (OS)(OT )Since the right-hand side (OS)(OT ) does not depend upon which line l ischosen, so long as it passes through O, then the left hand side, which givesthe power of l, is the same for all lines l that pass through O.(Case 3) O is any point ouside Γ and l is a secant line that intersects Γ atpoints Q and R.Construct line ←→ OA. Since one pont is inside Γ and one point is outside Γ,it intersects Γ at two points, which we denote by S and T .Then α = β (see figure 41.8) because they both intercept the same arc.Similarly δ = γ = 90 because the both interscept the same arc (and adiameter).Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.