2. From A, with radius A0, cut the circumference in D and E. 3. From D and E, draw lines through 0, cutting the circum ference in G and F, and the tangent in the point B. 4. Bisect the angle ABD (Pr. 4), and produce the line of bisection until it meets AC in H. 5. From 0, with radius OH, cut the lines EF and DG in the points K and L. 6. From H, K, and L, with radous HA, describe the three required circles, each of which will touch the other two, and the given outer circle 0. Problem 122. To inscribe four equal circles in a given square ABCD, touching each other, and one side only of the square. 1. Draw the diagonals AD, BC. With centres A, B, and C, and any radius, describe arcs at E and F. 2. From E and F, draw the diameters EG, FI. 3. The diagonals divide the square into four equal triangles, viz., AKB, BKD, CRD, and AKC. We have therefore only to describe a circle in each (Pr. 107). 4. In the triangle ARB the angle AKB is already bisected by KE; by bisecting one of the other angles, say KAB, by the line AL (Pr. 4), we obtain point L, the centre of one of the circles. M, N, O. Then with centres L, M, N, O, and radius Problem 123. To inscribe four equal circles within a given square ABCD, touching each other, and each circle also to touch two sides of the given square. 1. Draw the diagonals AD, BC. With centres A, B, and C, and any radius, describe arcs at E and F. 4. With centres M, N, P, and Q, and radius Mo, describe the four required circles within the given square ABCD. Problem 124. 1. Draw any two diagonals at right angles to each other, as AB, CD, intersecting each other in the centre 0. These divide the given octagon into four equal trapezia. 2. Find the centre of each, as E, F, G, H, and inscribe a circle in each trapezium (Pr. 110). The required four equal circles will then be inscribed in the given octagon. Problem 125. 1. Draw the diameters BC and DE at right angles to each other. 2. From B, C, D, and E, describe arcs cutting each other in F, G, H, K. 3. Join these points, and a square will be described about the circle A. 4. Draw the diagonals FK and HG. 5. Bisect the angle DGA (Pr. 4), and produce the line of bisection until it cuts DE in L. 6. From A, with radius AL, describe a circle cutting the lines BC and DE in M, N, O. 7. From centres L, M, N, 0, with radius LD, describe the four required circles within the given circle A. Problem 128. To inscribe five equal circles in a given circle A. 1. Divide the circumference into five equal parts, as in the case of inscribing a pentagon (Pr. 63). 2. From the centre A, draw lines through two divisions, as B and C, and produce them. 3. Bisect the angle BAC (Pr. 4), and draw AD, touching the circumference of the given circle in D. 4. At D, draw a tangent to the circle (Pr. 54), cutting AB and AC produced, and completing the triangle EAF. 5. Inscribe a circle in this triangle (Pr. 107), having its centre at G. 6. From A, with AG as radius, inscribe the circle GHKLM, cutting AE and AF in N and 0. 7. From A, with the line No as radius, cut the circum ference of the inner circle in H, K, L, M. 8. From those points, with radius DG, describe the remaining four circles within the given circle A, to complete the figure. |