Plane Geometry - Bruce E. Shapiro

SECTION 40. CIRCLES AND TRIANGLES 217By a similar argument, C ∗ G ∗ B.By AASHence△AEH ∼ = △AEF△BEG ∼ = △BEFEH = EF = EGLet r = EH; then points F , G, and H are equidistant from E and lie on acircle ΓC(E, r).Figure 40.3: Proof of the Inscribed Circle TheoremFurthermore, the sides of the triangle are tangent to the circle because thethree radii are perpendicular to the corresponding sides. Hence Γ is aninscribed circle.Definition 40.9 Let P 1 , P 2 , ...P n , n ≥ 3, be points such that no three ofpoints are collinear and that the segments P i P i+1 (including P n P 1 ) shareat most an endpoint. Then a polygon is the union of the segmentsP 1 P 2 ∪ P 2 P 3 ∪ · · · ∪ P n−1 P n ∪ P n P 1The points P i are called the vertices of the polygon and the segments arecalled the edges.Definition 40.10 A polygon is called a regular polygon if all of its sidesare congruent and all of its angles are congruent.Definition 40.11 A polygon is said to be inscribed in a circle if all of itsvertices lie on the circle.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.