PROPOSITION XIX. PROBLEM. 399. To inscribe in a given circle a regular polygon similar to a given regular polygon. _ DI CA Let A B C D, etc., be the given regular polygon, and C'D' E' the given circle. It is required to inscribe in C'D' E a regular polygon similar to A B C D, etc. From 0, the centre of the polygon A B C D, etc. draw 0 D and 0 C. draw O'C' and O'D', Draw C" D'. Then C" D' will be a side of the regular polygon required. For each polygon will have as many sides as the 20 =20') is contained times in 4 rt. As. .. the polygon C' D'E', etc. is similar to the polygon C D E, etc, § 372 (two regular polygons of the same number of sides are similar). Q. E. F. PROPOSITION XX. PROBLEM. 400. To circumscribe about a circle a regular polygon similar to a given inscribed regular polygon. Let H MRS, etc., be a given inscribed regular polygon. It is required to circumscribe a regular polygon similar to HMRS, etc. At the vertices H, M, R, etc., draw tangents to the O, intersecting each other at A, B, C, etc.. Then the polygon A B C D, etc. will be the regular polygon required. Since the polygon A B C D, etc. has the same number of sides as the polygon H MRS, etc., it is only necessary to prove that A B C D, etc. is a regular polygon. § 372 the A BHM, B MH, CMR, and CRM are equal, $ 209 (being measured by halves of equal arcs) ; .. the A BHM and C M R are equal, § 107 (having a side and two adjacent 1 of the one equal respectively to a side and two adjacent of the other). ..ZB=LC, (being homologous 5 of equal A ). .. the polygon A B C D, etc., is equiangular. (being homologous sides of equal isosceles S). .. the sides A B, BC, C D, etc. are equal, Ax. 6 and the polygon A B C D, etc. is equilateral. Therefore the circumscribed polygon is regular and similar to the given inscribed polygon. § 372 Q. E F. Ex. Let R denote the radius of a regular inscribed polygon, g the apothegm, a one side, A one angle, and C the angle at the centre; show that 1. In a regular inscribed triangle a = R V3, r = { R, A = 60°, C= 120°. 2. In an inscribed square a = RV2, n=1 RV2, A = 90°, C = 90°. 3. In a regular inscribed hexagon a = R, p = \ R V3, A= 120°, C = 60°. 4. In a regular inscribed decagon a = (V3_ ) go= IR V10 + 2 V5, A = 144°, C=36o. PROPOSITION XXI. PROBLEM. 401. To find the value of the chord of one-half an arc, in terms of the chord of the whole arc and the radius of the circle. D B Let A B be the chord of arc A B and A D the chord of one-half the arc A B. It is required to find the value of A D in terms of A B and R (radius). From D draw D H through the centre 0, and draw 0 A. HD is I to the chord A B at its middle point C, $ 60 (two points, 0 and D, equolly distant from the extremities, A and B, determine the position of a I to the middle point of A B). The 2 HA D is a rt. Z, § 204 (being inscribed in a sernicircle), .. A D = DH X DC, $ 289 (the square on one side of a rt. A is equal to the product of the hypoten use b? the adjacent segment made by the I let fall from the vertex of the rt. 2). Now and DH= 2 R, |