49. The lines joining the points of tangency of the escribed circles with the opposite vertices of the triangle ABC, are concurrent. [See Ex. 5.] 50. Deduce the Pythagorean Theorem (Prop. XI, Bk. IV) from Exercise 48. 51. Through a point P within an angle draw a line such that it and the parts of the sides that are intercepted shall contain a given area. [Construct parallelogram BDEF= B H required area (Ex. 38), DE passing through P. If HG is the required line, ▲ PIE = ▲ IFH + ▲ PDG. The ▲ are similar, DP, PE, and FH are homologous sides, and DP and PE are known.] 52. Is there any limit to the "given area" in Exercise 51 ? 707. DEFINITION. BOOK V A regular polygon is a polygon that is both equilateral and equiangular. PROPOSITION I. THEOREM 708. If the circumference of a circle is divided into three or more equal parts, the chords joining the successive points of division form a regular inscribed polygon; and tangents drawn at the points of division form a regular circumscribed polygon. B A х Ӧ Let the arcs AB, BC, etc., be equal. ... To Prove the polygon ABCD a regular inscribed polygon. [The proof is left to the student.] Let the arcs AB, BC, etc., be equal. To Prove the polygon xyz... a regular circumscribed polygon. Proof. [Draw the chords AB, BC, etc. Show that the A AуB, BzC, etc., are isosceles A and are equal in all respects.] 220 709. COROLLARY I. If at the middle points of the arcs subtended by the sides of a regular inscribed polygon, tangents to the circle are drawn, I. The circumscribed polygon formed is regular. II. Its sides are parallel to the sides of the inscribed polygon. M У E III. A line connecting the center of the circle with a vertex of the outer polygon passes through a vertex of the inner polygon. [yo bisects ZMON, consequently bisects arc MN, and therefore passes through B.] 710. COROLLARY II. If the arcs subtended by the sides of a regular inscribed polygon are bisected, and the points of division are joined with the extremities of the arcs, the polygon formed is a regular inscribed polygon of double the number of sides; and if at the extremities of the arcs and at their middle points tangents are drawn, the polygon formed is a regular circumscribed polygon of double the number of sides. 711. COROLLARY III. The area of a regular inscribed polygon is less than that of a regular inscribed polygon of double the number of sides; but the area of a regular circumscribed polygon is greater than that of a regular circumscribed polygon of double the number of sides. 712. EXERCISE. is regular. An equiangular polygon circumscribed about a circle 713. EXERCISE. An inscribed equiangular polygon is regular if the number of its sides is odd. 714. EXERCISE. A circumscribed equilateral polygon is regular if the number of its sides is odd. PROPOSITION II. THEOREM 715. A circle can be circumscribed about any regular polygon; and one can also be inscribed in it. I. To Prove that a circle can be circumscribed about it. Proof. Pass a circumference through three of the vertices, A, B, and C, and let o be its center. Draw the radii 0.4, OB, and Oc. Draw OD. Show that 21=Z Band Z3=Zc. Prove A OCB and OCD equal in all respects. Whence OD = OB. Therefore the circumference that passes through A, B, and C will also pass through D. Similarly, it can be shown that this circumference passes through the remaining vertices. Q.E.D. II. To Prove that a circle can be inscribed in the polygon. Proof. Describe a circle about the regular polygon AB... G. The sides AB, BC, etc., are all equal chords of this circle, and are equally distant from the center (?). With o as a center and this distance for a radius describe a circle. Show that AB, BC, etc., are tangent to this circle, which is, therefore, a circle inscribed in the regular polygon. Q.E.D. 716. DEFINITIONS. The common center of the circles that are inscribed in and circumscribed about a regular polygon, is called the center of the polygon. The angles formed by radii drawn from this center to the vertices of the polygon are called angles at the center. Each angle at the center is equal to 4 right angles divided by the number of sides in the polygon. A line drawn from the center of the polygon perpendicular to a side, is an apothem. The apothem of a regular polygon is equal to the radius of the inscribed circle. 717. EXERCISE. equilateral triangle? polygon of n sides? 718. EXERCISE. center is 30° ? 18° ? How many degrees in the angle at the center of an How many sides has the polygon whose angle at the 719. EXERCISE. In what regular polygon is the apothem one half the radius of the circumscribed circle ? 720. EXERCISE. In what regular polygon is the apothem one half the side of the polygon ? 721. EXERCISE. Show that an angle at the center of any regular polygon is equal to an exterior angle of the polygon. PROPOSITION III. THEOREM 722. Regular polygons of the same number of sides are similar. A a b с [Show that the polygons are mutually equiangular and have their homologous sides proportional.] |