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2. From A, with radius A0, cut the circumference in D
and E. 3. From D and E, draw lines through O, 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 O.
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, FH.
3. The diagonals divide the square into four equal triangles,
viz., AKB, BKD, CKD, 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.
5. With centre K, and radius KL, mark off the points
M, N, O. Then with centres L, M, N, O, and radius
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.
3. Join KG, GL, LH, and HK. Also join MN.
H 4. With centres M, N, P, and Q, and radius MO, describe
the four required circles within the given square ABCD.
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.
To inscribe four equal circles in a given circle A.
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.
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 O. 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.