Plane Geometry - Bruce E. Shapiro

218 SECTION 40. CIRCLES AND TRIANGLESTheorem 40.12 Let Γ be circle and P a point on Γ. Then for each n ≥ 3there is a regular polygon inscribed in Γ with a vertex at P .Proof. Let P 1 be defined on Γ such that m(∠P OP 1 ) = 360/n.Continue to construct points P 2 , P 3 , ...P n−1 such that m(∠P i OP i+1 ) =360/n.Then all of the triangles P i OP i+1 are congruent to one another.Hence all of the sides P i P i+1 are congruent to one another.Since each interior angle of P P 1 P 2 ...P n−1 P satisfiesm(∠P i−1 P i P i+1 ) = m(∠P i−1 P i O) + m(∠P i OP i+1 )= 180 − 360nthen each of the interior angles are congruent. Hence the polygon P P 1 P 2 ...P n−1 Pis regular.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012