Plane Geometry - Bruce E. Shapiro

SECTION 41. EUCLIDEAN CIRCLES 223Figure 41.4: Proof of Central Angle theorem when O lies on an edge of theinscribed angle.Henceγ = ɛ + ζ = 2β + 2δ = 2α(Case 3) O is neither on ∠P QR nor in its interior.Let S ∈ Γ such that Q ∗ O ∗ S.Either R is in the interior of ∠P QS or P is in the interior of angle ∠RQS.We can assume the first case (R ∈ interior(∠P QS)); if the second case istrue, exchange the labels of R and P .Then (see figure 41.6)α = β − δBy Case 1Henceɛ = 2δγ + ɛ = 2βγ + 2δ = 2βγ = 2(β − δ) = 2αCorollary 41.11 (Inscribed Angle Theorem) If two inscribed anglesintercept the same arc then they are congruent.Proof. This follows form the central angle theorem because if two anglesinscribe the same arc then they share the same corresponding central angle.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.