triangle. The point O will also be the centre of the circumscribing circle of radius O A.

70. In a given square to inscribe a circle.-Let ABCD (fig. 67) be the given square. Draw the diagonals A C, B D,

intersecting at O. Draw OE perpendicular to AB or parallel to AD, and the circle described from O as a centre, with OE as radius, will touch all the sides of the

square.

71. In any regular polygon to inscribe a circle.-Let

72. In a given square to inscribe four equal circles.

Let ABCD (fig 69) be the given square. Draw the diagonals AC, BD, intersecting in O. Draw EOG parallel to BC, and FOH parallel to A B. In each of the four squares into which the figure is divided inscribe a circle (70), and the four circles will touch each other, and also the sides of the given square.

Fig. 69.

73. In any given regular polygon to inscribe as many equal circles as the figure has sides.-Let A B, BC (fig. 68) be two adjacent sides of the given polygon ABCDE. Bisect the sides in H and G; and draw HO, GO at right angles to AB, BC, and let them meet at O. Join O B and bisect the angle OHB by the line HR, meeting OB in R; draw RS perpendicular to HB. Then a circle described

from R as a centre, with RS as radius, will touch the lines HB, BG, GO, and OH, and will be one of the inscribed circles. Proceeding in the same way at every corner of the figure, as many circles can be inscribed in it as the figure has sides or corners.

74. In a given circle to inscribe three equal circles.-Let O (fig. 70) be the centre, AO B one of the diameters of the

producing them to meet the Bisect the angle OG B by From O as a centre, and circle cutting the other

given circle. From A and B as centres, and with OA as radius, describe arcs cutting the circle in C, D, E, F. Draw the tangent (48) GBR perpendicular to AB, and also the diameters FOC, EOD, tangent at the points G and R. the line G H cutting AB in H. with OH as radius, describe a diameters in I and K; join HI, IK, KH; then circles drawn from H, I, and K as centres, with H B for radius, will touch each other, and also the given circle at the points B, E, and F.

Having drawn the diameters AOB, EOD, FOC, we can find the centres of the circles by measurement on a

scale; for if the radius OA of the given circle measures on any scale 1,000, then BH will be 464 and OH will be 536.

If it is required to draw three smaller circles touching the given one and the three inscribed circles, produce BA to L, making A L equal to HB. Draw L K and bisect it in M; let MN, the perpendicular to LK, cut the line A B in N; then a circle described from N as a centre, and with NA as radius, will touch two of the inscribed circles, and also the given circle at A. In the same way the other circles at C and D can be drawn with radii equal to A N.

Fig. 71.

75. To inscribe six equal circles in a given circle.-Let O be the centre, AOB (fig. 71) a diameter of the given circle. Draw the tangent (48) GAH, and also the diameters EOC, DOF, dividing the circle into six equal sectors as in the last problem (74). Bisect each

of the angles EOA, AOF by the lines OG, OH, meeting the tangent in G and H. Bisect the angle OG H by G K, cutting A O in K. From O as a centre, with

OK as radius, describe a circle cutting the diameters in the points, L, M, N, &c. Then circles drawn from the points K, L, M, &c., as centres, with KA as radius, will all touch the given circle and will touch each other. A circle drawn from O as a centre, and with radius OS equal to KA, will touch each of the inscribed circles. The radius of each of the inscribed circles in one-third that of the given circle.

76. To inscribe twelve equal circles in a given circle.

Let O (fig. 72) be the centre, AO the radius of the given

Fig. 72.

circle. Divide the circle into twelve equal sectors as in the last problem (75), and bisect one of those sectors by the radius OC; draw the tangent (48) BC at C perpendicular to OC, and bisect the angle CBO by the line BD, cutting OC in D. Bisect each of the other angles, and, from O as a centre, draw a circle cutting the radii in the Then circles drawn from the points D, E, F, &c., as centres, and with DC as radius, will touch each other and the circumference of the given circle.

Taking the radius OA as 1,000 on any scale, the length of the radius CD will be 206.

77. To inscribe four equal circles in a given circle.-Let

Fig. 73.

EOF, GOH (fig. 73) be two diameters of the given circle, intersecting each other in the centre O, at right angles to one another. Draw the tangent (48) AFB parallel to GH. Bisect the angles FOG, FOH by the lines OA, OB cutting the tangent in A and B. Let OA cut the circumference at P; on FO take FK equal to P-A, then the circle drawn

from K as a centre, with KF as radius, will touch the given circle at F, and also the lines OA, OB. Then from O as