radius O M, and draw ON making with O M the angle MON equal to the angle CBL (7). Draw the tangent SNR at right angles to the radius ON, and cutting QMR in the point R. Draw OP making with ON the angle PON equal to the angle K A C, and at P draw the tangent SPQ at right angles to OP, and cutting RMQ in the point Q. Then the triangle SRQ is similar to the triangle ABC, and is circumscribed about the given circle.

86. In a given circle to inscribe an equilateral triangle. -Let O be the centre (fig. 82) of the given circle, DOC a

diameter. From D as a centre, with DO as radius, describe an arc cutting the circle in A and B. Join A, B, C ; then ABC is an equilateral triangle inscribed in the circle. 87. About a given circle to circumscribe an equilateral triangle.-Let O be the centre (fig. 82), DOC a diameter of the given circle. Produce DC to F, making CF equal to the radius of the circle. From the point F draw FG E, FHK tangents (51) to the circle at G and H. Draw KDE the tangent (48) at D, and cutting the other tan

gents in the points E and K. Then KEF is an equilateral triangle circumscribed about the given circle.

88. In a given circle to inscribe a square.-Let O be the centre (fig. 67) of the given circle, EOF a diameter. Draw the diameter GOH at right angles to EOF, and draw the chords EH, HF, FG, GE. Then GEHF is a square inscribed in the given circle.

89. About a given circle to circumscribe a square.-Let O be the centre (fig. 67) of the given circle, EOF a diameter. Draw the diameter GOH at right angles to EOF; and draw AEB, DFC, tangents at E and F, parallel to GOH. Also draw AGD, CHB, tangents at G and H, parallel to EOF. Then ABCD is a square circumscribed about the given circle.

90. In a given circle to inscribe a regular octagon or figure of eight equal sides.-Let O be the centre (fig. 67) of the given circle, HOG, EOF two diameters at right angles to each other. Bisect the angles EOG, EOH by the diameters IOL, KOM, cutting the circle in I, K, L, M. Draw the chords IE, EK, KH, &c., and the inscribed octagon is completed.

91. About a given circle to circumscribe a regular octagon.-Let O be the centre (fig. 67) of the given circle, ABCD the circumscribed square (89). From A, B, C, and D as centres, with A O as radius, describe arcs cutting the sides of the square at P, Q, R, &c. Then the lines joining these points will form a circumscribed octagon.

Another method is to draw the diagonals A C, BD of the circumscribed square cutting the circle in I, K, L, M. Draw tangents PQ, RS, &c., at I, K, &c., parallel to the diagonals, and the octagon is completed.

92. In a given circle to inscribe a regular pentagon or

decagon. Let O be the centre (fig. 68) of the given circle, OF a radius. Construct the isosceles triangle OFL (25), having each angle at the base OFL, OLF, double the angle FOL at the vertex. Then FL is one side of an inscribed decagon or polygon of ten sides. Take LK equal to LF and draw the chord FK, which is one side of an inscribed pentagon or figure of five sides. By drawing arcs from F and K as centres, with FK as radius, cutting the circle at H and I, we obtain two more points on the pentagon; and drawing the diameter LOG, we find the fifth point G. Draw the chords FH, HG, GI, IK, and the pentagon is completed.

Bisect the angles FOH, HOG, GOI, IOK, by the lines O M, ON, &c., and the points L, F, M, H, N, &c., are the vertices of the inscribed decagon, and by drawing the chords FM, MH, &c., the decagon is completed.

93. About a given circle to circumscribe a regular pentagon or decagon.-Let O be the centre (fig. 68) of the given circle, FOP one of the diameters. Construct the isosceles triangle FOL, having each angle at its base FL double the angle at the vertex 0 (25). Draw the tangent AFE at right angles to OF, and produce OL to meet it at E; take FA equal to FE, and E A will be one side of the required pentagon. Draw the diameter LOG, and bisect the angles FOG, LOP by the diameter HOQ. Join AO and produce AO to meet the circle at I. At H, G, and I draw the tangents A H B, BGC, CID, each equal in length to A E; and join DE. Then ABCDE is the circumscribing pentagon or figure of five sides.

To construct the circumscribing decagon, or figure of ten sides, draw tangents at L, M, N, P, and Q on the circumference, cutting the sides of the circumscribed pentagon,

and the figure thus formed will be the circumscribing decagon.

94. In a given. circle to inscribe a regular hexagon, or figure of six equal sides.-Let O be the centre (fig. 83),

AOD the diameter of the given circle. From A and D as centres, and with AO as radius, describe arcs cutting the circle in B, C, E, and F. Draw the chords A B, BC, &c., and the inscribed hexagon ABCDEF is completed.

95. About a given circle to circumscribe a regular hexagon.-Let O be the centre (fig. 83) of the given circle, AOD a diameter. From A and B as centres, and with AO as radius, describe arcs cutting the circle in B, C, E, and F. Draw the tangents to the circle (48) at the points A, B, C, D, E, and F, and let them meet at the points H, K, L, M, N, and P; then the figure HKLMNP will be the circumscribed hexagon.

96. To inscribe in a given circle a regular dodecagon, or figure of twelve equal sides.-Let O be the centre (fig. 84) of the given circle, A OB, COD two diameters at right

angles to each other. From A, B, C, and D as centres, and with OA as radius, describe arcs cutting the circle in

E, F, G, H, &c. Draw the chords A E, EF, FC, &c., and the dodecagon is inscribed in the given circle.

97. About a given circle to circumscribe a regular dodecagon-Let O be the centre (fig. 83) of the given circle. Construct the circumscribing hexagon (95) HK L M N P, and draw the diagonals HM, KN, LP, cutting the circumference in a, b, c, d, e, f. Draw tangents at the points a, b, &c., intersecting the sides of the circumscribing hexagon in the points g, h, i, k, &c., and a dodecagon ghikl, &c., will be circumscribed about the circle.

98. In a given circle to inscribe a quindecagon, or figure of fifteen equal sides.-Let O be the centre (fig. 85) of the given circle, A O B a diameter. From A as a centre, and with A O as radius, describe an arc cutting the circumference in C and H. Draw the chords CH, HB, BC; then BCH is an inscribed equilateral triangle.

From C inscribe a regular pentagon (92) CDEFG in