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Essentials of Geometry

Essentials of Geometry


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1883 Excerpt: ...and XIII.) THEOREM IV. In any triangle, the bisector of an exterior angle divides the opposite side, externally, into segments proportional to the adjacent CHAPTER VIII. SIMILARITY OF POLYGONS. DEFINITIONS. 1. Similar Polygons are such as have the same form. Or, scientifically defined: Similar Polygons are those which are mutually equiangular, and whose corresponding sides are proportional. 2. The points, lines, or angles similarly situated, in similar polygons, are said to be homologous. THEOREM I. Triangles which are mutually equiangular, are similar. In the triangles ABC and D E F, let A = D, B = E, and C = F; then will they be similar. For, take A G = D E, A H C E H K, parallel to A B. The triangles A G H and D E F are equal throughout ( Ch. II, Th. IV); hence, a = c, and G H = E F; but b = c; hence, a = b; hence, G H and B C are parallel; hence, A B: A G:: A C: A H-Ch. VII, Th. I, Cor. I. But A G = D E, and AH = DF;.-. (1) AB: DE:: AC: DF. Since H K is parallel to A B; then, AC: AH:: BC: BK--Ch. VII, Th. I, Cor. I But BK = GH (Ch. II, Th. XXIII, Cor.), which, as we have seen, equals E F; hence, B K = E F. But A H is equal toDF; (2) AC: DF:: BC: EF. Combining (1) and (2), AB: DE:: AC: DF:: BC: EF. Therefore, the triangles, being mutually equiangular, and having their corresponding sides proportional, are similar. Q. K D. Cor. I.--Two triangles iye similar, when they have two angles of the one equal to two angles of the other, each to each. Cor. II.--In similar triangles, the corresponding or homologous sides lie opposite equal angles. This is evident from a simple inspection of the diagrams, --noting the relative position of the proportional sides. Y THEOREM II. Triangles which have their homologous sides proportional, are similar. In the triangles A


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Product Details
  • ISBN-13: 9781235259784
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 56
  • Spine Width: 3 mm
  • Width: 189 mm
  • ISBN-10: 1235259781
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 118 gr

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