Elements of geometry and trigonometry . le of the polygon ; so also are theangles B and b ; hence the triangles ABO, aho are similar, asare likewise the right angled triangles ADO, ado ; henceAB : ab : : AO : ao : : DO : do ; hence the perimeters of thepolygons are to each other as the radii AO, ao of the circum-scribed circles, and also, as the radii DO, do of the inscribedcircles. The surfaces of these polygons are to each other as thesquares of the homologous sides AB, ab ; they are thereforelikewise to each other as the s(juares of AO, r/o,the radii of thecircumscribed circles, or as the s
Elements of geometry and trigonometry . le of the polygon ; so also are theangles B and b ; hence the triangles ABO, aho are similar, asare likewise the right angled triangles ADO, ado ; henceAB : ab : : AO : ao : : DO : do ; hence the perimeters of thepolygons are to each other as the radii AO, ao of the circum-scribed circles, and also, as the radii DO, do of the inscribedcircles. The surfaces of these polygons are to each other as thesquares of the homologous sides AB, ab ; they are thereforelikewise to each other as the s(juares of AO, r/o,the radii of thecircumscribed circles, or as the squares of OD, of/, the radii ofthe inscribed circles. PROPOSITION XI. THEOREM. The (ti( ijj <[/(irs (tic ta Idch other as radiifand the areas are to each other as the squares of their radii. ^Wf| 120 GEOMETRY Let us designate the circumference of the circle whose radiusis CA by circ. CA ; and its area, by area CA : it is then to beshown that circ. CA : circ. OB : : CA : OB, and thatarea CA : area OB : : CA- : OB^.. Inscribe within the circles two reguJar polygons of the samenumber of sides. Then, whatever be the number of sides,their perimeters will be to each other as the radii CA and OB(Prop. X.). Now, if the arcs subtending the sides of the polygons be continually bisected, until the number of sides of thepolygons shall be indefinitely increased, the perimeters of thepolygons will become equal to the circumferences of the cir-cumscribe^ circles (Prop. VIII. Cor. 2.), and we shall havecirc. CA : circ. OB : : CA : OB. Again, the areas of the inscribed polygons are to each otheras CA^ to OB- (Prop. X.). But when the number of sides ofthe polygons is indefinitely increased, the areas of the polygonsbecome equal to the areas of the circles, each to each, ( Cor. 1.) ; hence we shall have area CA : area OB : : CA^ : OB^. Cor. The similar arcs AB,DE are to each other as theirradii AC, DO ; and the similarsectors ACB, DOE, are to eachother as the squar
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