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" Two similar polygons are composed of the same number of triangles, similar each to each, and similarly situated. "
Elements of Geometry and Trigonometry: With Notes - Page 74
by Adrien Marie Legendre - 1830 - 316 pages
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Elements of Geometry: And the First Principles of Modern Geometry

William Henry Harrison Phillips - Geometry - 1878 - 236 pages
...be shown that the remaining triangles are similar each to each. XXVIII. Theorem. Conversely, if tu'o polygons are composed of the same number of triangles similar each to each, and similarly situated, the polygons are similar. . HYPOTII. In the two polygons, ABCDE and abcde, A ABC ~ abc, ACD ~ acd,...
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Elements of Plane Geometry, Part 1

Thomas Hunter - Geometry, Plane - 1878 - 142 pages
...equal to ADxDC (Prop. XIX.); and, by substitution, ABxBC=ADxDC+BD 2 . PROPOSITION XXIII.—THEOREM. Two similar polygons are composed of the same number of triangles, similar to each other, and similarly situated* Let ABCDE and FGKLM be two similar polygons; from any angle,...
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Elements of Geometry, After Legendre, with a Selection of Geometrical ...

Charles Scott Venable - 1881 - 380 pages
...triangles be shown to be similar, whatever were the number of sides of the proposed polygons ; therefore, two similar polygons are composed of the same number of triangles, similar and similarly situated. SCHOLIUM. The converse proposition is equally true : If two polygons are composed...
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Elements of Plane Geometry

Franklin Ibach - Geometry - 1882 - 208 pages
...:: HK : FG :: CK : CG ELEMENTS OF PLANE GEOMETRY. THEOREM XX. 277. Two polygons are similar if they are composed of the same number of triangles similar each to each and similarly placed. Let the As ABE, EBD, and DBC, of which the polygon ABСDE is composed, be respectively similar...
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A Geometry for Beginners

George Anthony Hill - 1886 - 368 pages
...is similar to AB CD E. Hence we arrive at the following useful theorem : — Theorem I. — If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. The converse theorem is also true. State it. If each side of a polygon...
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Chauvenet's Treatise on Elementary Geometry

William Chauvenet, William Elwood Byerly - Geometry - 1887 - 331 pages
...PROPOSITION V. Two triangles are similar when their homologous sides are proportional. PROPOSITION VI. If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. PROPOSITION VII. Two similar polygons may be decomposed into the...
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A Text-book of Geometry

George Albert Wentworth - Geometry - 1888 - 264 pages
...coincides with F. :. EF prolonged passes through O. SIMILAR POLYGONS. PROPOSITION XII. THEOREM. 331. If two polygons are composed of the same number of triangles, similar each to each, and similarly placed, the polygons are similar. E BCW l!* In the two polygons ABCDK and A'B ' <yD>E', let the triangles...
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The Elements of Plane and Solid Geometry ...

Edward Albert Bowser - Geometry - 1890 - 414 pages
...the lengths of D'E' and E'C'. BOOK III.— SIMILAR FIGURES. Proposition 2 1 . Theorem. 320. If two polygons are composed of the same number of triangles similar each to each, and similarly placed, the polygons are similar. Hyp. Let the AS ABC, ACD, ADE of the polygon ABCDE be similar respectively...
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Plane and Solid Geometry

Seth Thayer Stewart - Geometry, Modern - 1891 - 422 pages
...polygons may be divided into the same number of similar triangles, similarly placed. PROP. XIII. Polygons composed of the same number of triangles, similar each to each, and similarly placed, are similar. PROP. XIV. The perimeters of similar polygons are proportional to any homologous...
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Elementary Geometry

William Chauvenet - 1893 - 340 pages
...PROPOSITION V. Two triangles are similar when their homologous sides are proportional. PROPOSITION VI. If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar. PROPOSITION VII. Two similar polygons may be decomposed into the...
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