Elements of geometry and trigonometry . The point O, the common centre of the in-scribed and circumscribed circles, may also be regarded as thecentre of the polygon ; and upon this principle the angle AOBis called the angle at the centre, being formed by two radiidrawn to the extremities of the same side AB. Since all the chords AB, BC, CD, &c. are equal, all the an-gles at the centre must evidently be equal likewise ; and there-fore the value of each v/ill be found by dividing four right an-gles by the number of sides of the polygon. BOOK V. Ill Scholium 2. To inscribe a regu-lar polygon of a
Elements of geometry and trigonometry . The point O, the common centre of the in-scribed and circumscribed circles, may also be regarded as thecentre of the polygon ; and upon this principle the angle AOBis called the angle at the centre, being formed by two radiidrawn to the extremities of the same side AB. Since all the chords AB, BC, CD, &c. are equal, all the an-gles at the centre must evidently be equal likewise ; and there-fore the value of each v/ill be found by dividing four right an-gles by the number of sides of the polygon. BOOK V. Ill Scholium 2. To inscribe a regu-lar polygon of a certain number ofsides in a given circle, we have onlyto divide the cireumterence into asmany equal parts as the p<jlyg<»nlias sides : tor the arcs being chords AU, BC, CD, &c. willalso be equal ; hence likewise thetriangles AOB, BOC, (X)I), mustbe equal, because siiies are equal each to each ; hence allthe angles ABC, BCD, CDE, «Sec. will be equal ; hence thefigure ABCDEII, will be a regular j) PROPOSITION III. PROBLEM. To inscribe a square in a givcji circle. Draw two diometers AC, BD, cut-ling each other at right angles : jointheir extremities A, B, C, 1): the figureAB(JD will be a square. For the an-gles AOB, BOC, &c. being equal, the Achords AB, BC, 6lc. are also equal :and the angles ABC, BCD, &c. beingin semicircles, are right angles. SchoJiujn. Since the triangle BCO is riglit angled and , we have BC : liO : : V2 ; 1 (Book IV. Prop. 4.); iiencc the side of the inscribed square is to the radius^(IS the square root of2f is to unity. PROPOSITION IV. PROnLExM.
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
