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a centre, with OK as radius, describe a circle cutting the diameters in L, M, N; the circles drawn from these points as centres, and with FK or AP as radius, will be inscribed in the given circle.

The point K can also be found by bisecting the angle OAB by the line A K.

If the radius of the given circle is 1,000 on any scale, then the length of AP or KF, the radius of each of the inscribed circles, will be 414, and that of OK 586, on the same scale.

In order to describe the small circles touching the given circle and the inscribed ones, draw KTR parallel to G.H, and cutting AO in R. A circle drawn from R as a centre, with RP as radius, will touch two of the inscribed circles, and also the given one at P. Also a circle drawn from O as a centre, with radius OS equal to RP, will touch all the inscribed circles.

78. To inscribe eight equal circles in a given circle.-Let O (fig. 74) be the centre, OC the radius of the given circle. Divide it, as in the last problem,

(77) into eight equal sectors, and let AOB be one of these sectors, the angle AOB being bisected by OC. Draw the tangent CT (48) perpendicular to OC and meeting OB in T. Bisect the angle OTC by the line DT, cutting OC in D. Then a circle described from D as a centre,

Fig. 74.

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with DC as radius, will touch the given circle at C, and also the lines AO, BO; this will be one of the required circles. From O as a centre, and with OD as radius,

describe a circle cutting all the other diameters in E, F, G, &c., which will be the centres of the other inscribed circles, their radius being equal to DC.

If OC is represented on scale by 1,000, then the length of CD will be 277, and that of OD 723.

79. In a given circle to inscribe five equal circles.-Let O (fig. 75) be the centre, BOA the diameter of the given

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Fig. 75.

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circle. Construct the isosceles triangle BOK (25), having each angle OKB, OBK at the base double the angle BOK at the vertex. Draw the tangent BT at right angles to BO, and produce OK to meet it at T. Bisect the angle OTB by the line TM, cutting OB in M. Then a circle described from M

as a centre, with M B as radius, will be one of the inscribed circles required. Take KG, GJ, JF, &c., each equal to KB, and draw the diameters through the points G, J, &c. From O as a centre, with OM as radius, describe a circle cutting these diameters in the points N, P, Q, R, which will be the centres of the other inscribed circles.

If the length OA of the radius of the given circle is represented on any scale by 1,000, then that of each inscribed circle is 370, and of O M 630.

80. In a given circle to inscribe ten equal circles.-Let O (fig. 76) be the centre and OA the radius of the given circle. Divide the circle into ten equal sectors, as in the last problem (79), and let AOB be one of those sectors. Bisect the angle AOB by the radius OC, and draw the tangent CT perpendicular thereto. Bisect the angle OTC

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by the line TD, cutting OC in D; then the circle drawn from D as a centre, with DC for radius, will be one of the inscribed circles required. Το

Fig. 76.

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is 1,000 on any scale, then that of each inscribed circle will be 236, and the length of OD is 794.

81. In a given circle to inscribe (approximately) seven equal circles.-The method of solving this problem is only an approximation, but sufficiently

true for most practical purposes.

Let O (fig. 77) be the centre,

AOB a diameter of the given circle. Draw DOC at right angles to AOB and produce it to X. Divide the radius into four equal parts and measure CX equal to three of those parts. Divide AB into seven

equal parts at a, b, c, d, e, f. Through b draw Xb E, meet

ing the circle at E. Draw the

Fig. 77.

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radius OE and bisect the angle EO A by the line HO; draw TH the tangent at Ḥ perpendicular to HO, and produce OE to meet it at T. Bisect the angle ATO by the line

TI, cutting OH in I; then a circle described from I as a centre, with IH as radius, will be one of the inscribed circles required. To find the rest, take HP, PF, F B, &c., each equal to A E, and draw the diameters through H, P, F, &c. From O as a centre, and with OI as radius, describe a circle cutting the diameters in the points S, U, &c., which will be the centres of the other inscribed circles.

If the radius of the given circle is 1,000 on any scale, then that of each inscribed circle will be 303, and the length of the radius OI will be 697, on the same scale. 82. To circumscribe a circle about a given triangle.

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83. To circumscribe a circle about any given regular polygon.-Let FHGIK (fig. 68) be the given polygon. Bisect the two adjacent angles KFH, FHG by the lines FO, HO, meeting at O. Then the circle described from O as a centre, with OF as radius, will pass through all the

vertices of the polygon.

INSCRIBED AND CIRCUMSCRIBED POLYGONS.

84. In a given circle to inscribe a triangle similar to a given triangle.--Let ABC (fig. 79) be the given triangle,

O the centre, and DG the diameter (fig. 80) of the given

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circle.

Draw GE making with the diameter the angle DGE equal to the angle A CB. Draw the chord E D, and draw DF making the angle EDF equal to the angle B A C. Join EF, then the triangle DEF is similar to the given triangle, and is inscribed in the given circle.

85. About a given circle to circumscribe a triangle similar to a given triangle.-Let ABC (fig. 79) be the given triangle, and let O be the centre (fig. 81) of the given circle, Fig. 81.

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OM its radius. Produce A B to K and L; at any point M draw the tangent to the circle QMR at right angles to the

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