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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

Adrien Marie Legendre - Geometry - 1819 - 208 pages
...proportion AC : DF : : AC : DF, we shall hare AB X AC; DE X DF : : AC : DF. Hence ABC : DEF : : AC : DF. Therefore two similar triangles ABC, DEF, are to each...polygons are composed of the same number of triangles, which are similar to each other and similarly disposed. Demonstration. In the polygon ABCDE (Jig. 129)...
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Elements of Geometry

Adrien Marie Legendre - 1825 - 574 pages
...identical proportion. AC:DF::AC:DF, we shall have AB x AC : DE x DF : : AC : DF. Hence ABC:DEF::TC:DF. Therefore two similar triangles ABC, DEF, are to each...squares of any other two homologous sides. THEOREM. 21 9. Two similar polygons are composed of the same number of triangles, which are similar to each...
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Elements of Geometry

Adrien Marie Legendre - Geometry - 1825 - 280 pages
...proportion. AC : DF : : AC : DF, we shall have AB x AC : DE x DF : : AC : DF. Hence ABC: DEF:: AC: DF. Therefore two similar triangles ABC, DEF, are to each...squares of any other two homologous sides. THEOREM. 21 9. Two similar polygons are composed of the same number of triangles, which are similar to each...
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Elements of Geometry

Adrien Marie Legendre - Geometry - 1825 - 224 pages
...identical proportion. AC:DF::AC: DF, we shall have Hence AB x AC : DE x DF : : AC : DF. ABC : DEF ::AC: DF. Therefore two similar triangles ABC, DEF, are to each...squares of any other two homologous sides. THEOREM. 21 9. Two similar polygons are composed of the same number of triangles, which are similar to each...
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Elements of Geometry

Adrien Marie Legendre - Geometry - 1825 - 224 pages
...identical proportion. AC:DF::AC:DF, we shaft have AB X AC : DE X DF : : AC : DF. Hence ABC : DEF ::AC: DF. Therefore two similar triangles ABC, DEF, are to each other as the squares of the homologous sides AC, OF, or as the squares of any other two homologous sides THEOREM. "/(. 219. Two similar polygons are...
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Elements of Arithmetic, Algebra, and Geometry

George Lees - 1826
...lines be cut by parallel lines, they are cut in the same ratio. 2. If, on two bases, there be placed the same number of triangles, similar each to each, and similarly situated, the polygons formed by joining the vertices of the triangles will be similar. 3. A straight line, drawn...
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Elements of Geometry: With Notes

John Radford Young - Euclid's Elements - 1827 - 208 pages
...number of triangles, similar each to each, and similarly situated ; and, conversely, polygons which are composed of the same number of triangles, similar each to each, and similarly situated, are themselves similar. ELEMENTS OP GEOMETRY. while the sides containing these angles are proportional...
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Elements of Geometry: With Notes

John Radford Young - Euclid's Elements - 1827 - 208 pages
...that is, the triangles are similar. PROPOSITION XIX. THEOREM. Similar polygons may be divided into the same number of triangles, similar each to each, and similarly situated ; and, conversely, polygons which are composed of the same number of triangles, similar each to each,...
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Elements of Geometry and Trigonometry

Adrien Marie Legendre - Geometry - 1836 - 359 pages
...remaining triangles are similar, whatever be the number of sides in the polygons proposed : . therefore two similar polygons are composed of the same number of triangles, similar, and similarly situated. Scholium. The converse of the proposition is equally true : If two polygons...
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Elements of Geometry and Trigonometry

Adrien Marie Legendre - Geometry - 1837 - 359 pages
...: : AC : DF, there will result AB.AC : DE.DF : : AC2 : DP. Consequently, " ABC : DEF : : AC2 : DF. Therefore, two similar triangles ABC, DEF, are to each other as the squares described on their homologous sides AC, DF, or as the squares of any other two homologous sides. PROPOSITION...
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