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" The area of a regular polygon inscribed in a circle is a geometric mean between the areas of an inscribed and a circumscribed regular polygon of half the number of sides. "
Elementary Trigonometry - Page 198
by James Hamblin Smith - 1877 - 228 pages
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An Elementary Treatise on the Geometrical and Algebraical Investigation of ...

Daniel Cresswell - Euclid's Elements - 1817 - 454 pages
...(LXII.) To describe a polygon, similar to a given polygon, and having a given ratio to it. (LXIII.) Any regular polygon, inscribed in a circle, is a mean proportional between the inscribed and circumscribed regular polygons of half the number of sides. (LXIV.) If from two points...
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A Supplement to the Elements of Euclid

Daniel Cresswell - Geometry - 1819 - 446 pages
...from E. 20. 6., that it will have to the given polygon the given ratio. PROP. LXVI. 75. THEOREM. > Any regular polygon, inscribed in a circle, is a mean proportional between the inscribed and circumscribed regular polygons of half the number of sides. Let BGFA be a polygon inscribed...
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Elements of Geometry, Containing the First Six Books of Euclid

Euclid - 1826 - 234 pages
...the first to the similar and similarly described rectilineal figure upon the second. Deduction. Any regular polygon inscribed in a circle, is a mean proportional between the inscribed and circumscribed regular polygon of half the number of sides. PROPOSITION XXI. THEOREM....
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Euclid's Elements of geometry, transl. To which are added, algebraic ...

Euclides - 1826 - 226 pages
...rectilineal figures are to one another in the duplicate ratio of their homologous sides. QEF Deduction. Any regular polygon inscribed in a circle, is a mean proportional between the inscribed and circumscribed regular polygon of half the number of sides. PROPOSITION XXI. » THEOREM....
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Problems in the Different Branches of Philosophy: Adapted to the Course of ...

Miles Bland - Astronomy - 1830 - 394 pages
...also the areas of regular polygons of a given number of sides, both inscribed and circumscribed. 73. The area of a regular polygon inscribed in a circle is a geometric mean between the areas of an " inscribed and circumscribed regular polygon of half the number...
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Mathematical Problems and Examples, Arranged According to Subjects, from the ...

Mathematics - 1836 - 366 pages
...opposite, having given B, a and the area of the triangle, find the remaining sides and angles. 25. Prove that the area of a regular polygon inscribed in a circle, is a geometric mean between the areas of an inscribed and of a circumscribed regular polygon of half the...
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A collection of examples in pure and mixed mathematics, with hints and ...

Alfred Wrigley - 1845 - 222 pages
...hypothenuse. 174. In the figure of Euclid, Book iv., prop. 10; compare the areas of the two circles. 175. The area of a regular polygon inscribed in a circle...circumscribed regular polygon of half the number of sides. 176. In a regular polygon of n sides, if each side =2a ; find the sum of the radii of the circumscribed...
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A collection of problems and examples adapted to the 'Elementary course of ...

Harvey Goodwin (bp. of Carlisle.) - Mathematics - 1847 - 136 pages
...the inscribed square is an harmonica) mean between the sides containing the right angle. 19. Prove that the area of a regular polygon inscribed in a circle is a geometrical mean between the areas of an inscribed and of a circumscribed polygon of half the number...
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A Treatise on Plane and Spherical Trigonometry ...

Thomas Grainger Hall - Trigonometry - 1848 - 192 pages
...the triangle ABO, OA* + ОБ2 + ОС1 = ab + ас + be - 12 Er. (13.) The area of a regular hexagon inscribed in a circle is a mean proportional between the areas of an inscribed and circumscribed equilateral triangle. (14.) The square of the side of a pentagon inscribed in a circle...
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The Elements of Plane Trigonometry

James Hann - Plane trigonometry - 1854 - 140 pages
...12ßr = ai + ac+ bc-l2Rr; .: АО' + OB' +OC' = ab + ac + bc- IZRr. (] 0) The area of a regular hexagon inscribed in a circle is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles. л г г. пт* . Sir Area of hexagon = —...
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