Sidebilder
PDF
ePub

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.

[graphic][subsumed][subsumed][ocr errors][subsumed][ocr errors][subsumed]

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.

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, 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).

[blocks in formation]

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
, describe the four required circles in the given square
ABCD.

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.
2. From E and F, draw the diameters GH and KL.

3. Join KG, GL, LH, and HK. Also join MN.

[blocks in formation]

с

H 4. With centres M, N, P, and Q, and radius MO, describe

the four required circles within the given square ABCD.

Problem 124.
To inscribe four equal circles in a given octagon.

С

[blocks in formation]

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.

To inscribe four equal circles in a given circle A.

D

ВІ

C

N

H
E

K
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).

F

• M:

K

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.

« ForrigeFortsett »