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" SIMILAR triangles are to one another in the duplicate ratio of their homologous sides. "
The Elements of Euclid; viz. the first six books, together with the eleventh ... - Page 167
by Euclides - 1814
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Annual Report of the Commissioners ..., Volume 40

1874
...Explain the term duplicate ratio, and illustrate its meaning as you would to a class. (b.) Prove that similar triangles are to one another in the duplicate ratio of their homologous sides. 2. Describe a circle which will pass through a given point, and touch a given circle in a given point....
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Rudimentary treatise on mensuration and measuring

Thomas Baker (C.E.) - 1874
...then, the triangle ABC is to the triangle AED as the square of A. B is to the square of AE : that is, similar triangles are to one another in the duplicate ratio of their homologous sides. (Euc. VI. 19.) THEOREM VIII. All similar figures are to one another as the squares of their homologous,...
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Euclidian Geometry

Francis Cuthbertson - Euclid's Elements - 1874 - 349 pages
...ratio of AB to AH. Hence, proceeding as in Proposition VIII., it may be proved that Similar polygons are to one another in the duplicate ratio of their homologous sides. PROPOSITION (r). If four straight lines are proportional, the duplicate ratio of the first two is the...
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Plane and solid geometry, by G.E. Webster and A. Gardiner

George E. Webster - 1874
...area is possessed by the figure which has the largest number of sides. (7) Similar rectilineal figures are to one another in the duplicate* ratio of their homologous^ sides. (8) If three straight lines oe proportionals, as the first quantity is to the third quantity, so is...
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Elements of Euclid [selections from book 1-6] adapted to modern methods in ...

Euclides - 1874
...homologous sides. Cor. 2. — In like manner it may be proved that any similar figures of four sides are to one another in the duplicate ratio of their homologous sides, and the same has been proved of triangles (VI. 14); therefore, universally, similar rectilineal figures...
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Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...

Robert Potts - Geometry, Plane - 1876 - 403 pages
...angles proportionals, they are similar to one another. In the same manner a rectilineal figure of six sides may be described upon a given straight line similar to one given, and so on. QEF PROPOSITION XIX. THEOREM. Similar trianyla are to one another in the duplicate ratio of their homologous...
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Elements of geometry, containing books i. to vi.and portions of books xi ...

Euclides, James Hamblin Smith - Geometry - 1876 - 349 pages
...more sides may be described, on a given line, similar to a given fig. QEF PROPOSITION XIX. THEOREM. Similar triangles are to one another in the duplicate ratio of their homologous sides. Let ABC,DEF be similar AS, having / s at A, B, C= L s at D, E, F respectively, so that BC and EF are homologous...
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Modern geometry [ed.] with an appendix by W.B. Jack

Richard Wormell - 1876
...circumscribed circles, or their proportionals the radii of the inscribed circles. THEOREM LXXXV. (¿.) Similar triangles are to one another in the duplicate ratio of their homologous sides. Let А В С, DEF be similar triangles having Z. B = ¿EandAB:DE = BC:EFso that В С and E^F are homologous...
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SYLLABUS OF PLANE GEOMETRY

1876
...to one another the ratio compounded of the ratios of their bases and of their altitudes. THEOR. 15. Similar triangles are to one another in the duplicate ratio of their homologous sides. THEOR. 16. The areas of similar rectilineal figures are to one another in the duplicate ratio of their...
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Elements of Geometry Containing Books I. to VI. and Portions of Books XI ...

James Hamblin Smith - Geometry - 1876
...described, on a given line, similar to a given fig. QEF PROPOSITION XIX. THEOREM. Similar triangles an to one another in the duplicate ratio of their homologous sides. Let ABC, DBF be similar A s, having / s at A, B, C= L s at D, E, F respectively, so that BC and EF are homologous...
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