Elements of geometry and trigonometry . s, and are consequently similar (Prop. XX.). In the samemanner it might be shown that all the remaining triangles aresimilar, whatever be the number of sides in the polygons pro-posed : therefore two similar polygons are composed of thesame number of triangles, similar, and similarly situated. Scholium, The converse of the proposition is equally true :Jf two polygons are composed of the same number of trianglessimilar and siinilarly situated, those two polygons will be similar. For, the similarity of the respective triangles will give theangles, ABC=FGH,
Elements of geometry and trigonometry . s, and are consequently similar (Prop. XX.). In the samemanner it might be shown that all the remaining triangles aresimilar, whatever be the number of sides in the polygons pro-posed : therefore two similar polygons are composed of thesame number of triangles, similar, and similarly situated. Scholium, The converse of the proposition is equally true :Jf two polygons are composed of the same number of trianglessimilar and siinilarly situated, those two polygons will be similar. For, the similarity of the respective triangles will give theangles, ABC=FGH, BCA-GHF, ACD = FHI : hence BCDn=GHI, likewise CDE=HIK, &c. Moreover we shall haveAB : FG : : BC : GH : : AC : FH : : CD : HI,&c.; hencethe two polygons have their angles equal and their sides pro-jwrtional ; consequently they are similar. PROPOSITION XXVII. THEOREM. The contours or perimeters of similar polygons are to each oiJieras the homologous sides : and the areas are to each other astJie squares described on those BOOK IV. P3 First. Since, by thenature of similar figures,we have AB : FG : :EC : Gil : : CD : HI,&.C. we conclude fromthis series of equal ratiosthat the sum of the ante-cedents AB4-BC-:-CD,«Sec, which makes up the perimeter of the first polygon, is tothe sum of the consequents FG + GII + HI, &c., which makesup the perimeter of the second polygon, as any one antecedentis to its consequent ; and therefore, as the side AB is to its cor-responding side FG (Book II. Prop. X.). Secondly. Since the triangles ABC, FGH are similar, weshall have the triangle ABC : FGH : : AC- : FH^ () ; and in like manner, from the similar triangles ACD,FHI, we shall have ACD : FHI^: : AC^ : FH-; therefore, byreason of tiie common ratio, AC^ : FH-, we have ABC : FGH : : ACD : the same mode of reasoning, we should find ACD : FHI : : ADE : FIK;and so on, if there were more triangles. And from this seriesof equal ratios, we conclude that tJie sum of the anteceden
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