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Describe it, and from C, with AB as a distance, set off on it the points D, E, join CD, DE, EB, and ACDEB is the required pentagon. If the pentagon has to be inscribed in a given circle, then from its centre which will be the centre of the pentagon draw any radius as SA (Fig. 67) at S, draw a line making with SA an angle of 72, and cutting the circle in B, join A and B, then AB is one side of the required pentagon ; set off the distance AB from A or B round the circle, and it will give points C, D, E ; join ACDEB, and the pentagon is constructed in the given circle. Problem 20 (Fig. 68). To construct a regular Jiexagon with a given length of side. Here 360 -=- 6 equals 60, and 180 - 60 = 120. Let AB be the given side, produce it, and draw AC, making with AB produced an angle of 60 ; make AC equal to AB, bisect them by perpendiculars intersecting in S, which is the centre of the circumscribing circle ; describe it, and set off the distance AB round it from C, in points D, E, F, join CD, DE, EF, FB, and the required hexagon is constructed. If a hexagon has to be inscribed in a given circle, the central angle will be 60 ; this angle laid down with the centre of the circle as the angular point will give A, B (Fig. 68), points in the circle, and the line joining them will be a side of the hexagon ; step this length round the circle in points C, D, E, F, join AC, CD, etc., and the required hexagon is inscribed in the given circle. As the side of a hexagon is the chord of an arc of 60, and is equal to the radius of the circumscribing circle, that radius set off round the circle will divide it into six equal parts, and if the points of division be joined by right lines they would form the inscribed hexagon as before.