Problem 51.

To describe a circle of a given radius A, touching any two given circles B and C, tangentially.

1. Draw a line of indefinite length through B and C.

2. Make DG and FE each equal to 4.

H
:K

A

3. With B as centre, and radius BG, describe the arc GH. 4. With C as centre, and radius CE, describe arc EK. With K as centre, and radius A, describe the required circle, which shall be tangential to the given circles B and C.

Problem 52.

To describe three circles having any given radii A, B, and C, each circle being tangential to the other two.

1. Take any point D as centre, and with line A as radius, describe a circle, and produce the radius DE outwards indefinitely.

2. From E, with line B as radius, cut DE produced in F; and from F, with the same radius, describe the second circle.

3. Draw any other radius to each circle as DH, FG, and produce them.

D

4. On the produced radii, set off HK, and GL, equal to C.

E

A

B

C

5. From points D and F, with DK and FL as radii respectively, describe arcs cutting each other in O.

6. From 0 as centre, with line C as radius, describe the remaining circle. Then F, D, O shall be the three required circles.

Problem 53.

To describe a series of circles in succession, tangential to two given converging lines AB and CD.

1. Bisect the angle made by the two given converging lines (Pr. 12), and take any point E in the line of bisection.

2. From E, draw a perpendicular EF to one of the given lines AB (Pr. 3).

3. Then E is the centre, and EF the radius of the first circle, which cuts the line of bisection in G.

4. From G, draw a line perpendicular to EG (Pr. 2), meeting AB in H.

5. From H, set off HK equal to HF, and draw a line from K, perpendicular to AB, or parallel to EF (Pr. 8), meeting the line of bisection in L.

6. Then L is the centre, and LK or LG the radius of the second circle, which cuts the line of bisection in O.

NOTE.

By the same construction other circles may be described

either towards A, C, or B, D.

Problem 54.

To draw a tangent to a given circle A at a given point of contact B in the circumference.

1. Find the centre of the circle A (Pr. 45), and from B draw a radius BA.

B

2. From B draw a line perpendicular to AB (Pr 2), and produce it both ways towards C and D. Then CD is the required tangent to the given circle A.

Problem 55.

To draw a tangent to a given circle A, from a given point B outside the circumference.

1. Find the centre of the circle A (Pr. 45), and draw a line from B to the centre A.

B

2. Bisect AB in the point C (Pr. 1), and describe the circle of which AB is the diameter, and cutting the given circle in the required points of contact D and E.

3. Join BD or BE, and either of these lines produced beyond D or E is the required tangent to the given circle A.

Problem 56.

To draw a tangent on the outside of two given equal circles A and B, placed apart.

1. Join the centres A and B of the given circles.

E

O O

2. At A and B, draw lines AC, BD at right angles to AB, meeting the circumferences in C and D (Pr. 2), the points of contact.

3. Draw the required tangent EF through the points C and D.

Problem 57.

To draw a tangent between two given equal circles A and B, placed apart.

A

F

1. Join the centres A and B of the given circles.

2. Bisect the line AB (Pr. 1) in the point C; and from C, draw a tangent to the circle A (Pr. 55) one point of contact being at D.

3. Produce the line CD both ways to E and F. Then EF will be the required tangent.

Problem 58.

To draw a tangent to any given point of contact A, in the given arc of a circle AB, when the centre cannot be obtained.

B

1. Draw the chord AB, and bisect it in C (Pr. 1).

2. From the point C, erect a perpendicular CD to AB (Pr. 2).

3. Join DA, and make the angle DAE equal to the angle

DAC (Pr. 10).

4. Produce AE, and it is the required tangent.