Let a circle be described with centre A, and the tangent at any point C be drawn; and let, with centre B, in AC produced, and radius BC, another circle be described. Then both circles have CD for tangent. Both touching the same line at the same point, they are

said to touch one another, in this case externally. Evidently, whether circles touch internally or externally, the straight line joining their centres passes through the point of contact.

Describe circles of radii 34 and 56 millimetres to touch (1) externally, (2) internally.

Construct a series of circles of radii 20, 17, 14, 11,

millimetres, their centres being in the same straight line, and each circle touching the preceding (and succeeding) externally.

Describe circles of radii as in preceding, but each circle touching the others at the same point, internally.

Two circles of radii 30 and 40 millimetres touch one another externally. Describe a circle of radius 20, to touch both of them externally. (This involves the construction of a triangle with sides 70, 60 and 50 millimetres.)

Make the same construction as in the preceding question, when the first two circles have radii 25 and 35, and the third a radius of 15 millimetres.

The sides of a triangle are 75, 60 and 45 millimetres. With the angular points of this triangle as centres, describe three circles with radii 15, 30 and 45 millimetres, so that each may touch the other two.

When the sides of the triangle are 100, 75 and 65 millimetres, discover the circles whose radii are such that in like manner each will touch the other two, the angular points of the triangle being centres of the circles.

Exercises.

1. Describe a circle of radius 40 millimetres; draw two diameters at right angles to one another; and draw tangents at ends of the diameters, and produce them so that they intersect. What do you observe as to lengths of tangents? What angles do they make with one another? Apply tests with dividers and set-square.

2. Describe a circle of radius 11⁄2 in.; draw diameters at intervals of 60°; and draw tangents at ends of diameters. What do you observe as to lengths of tangents? What angles do they make with one another? Apply tests.

3. Describe a circle of radius 14 in.; draw any line in plane of paper; draw a tangent parallel to this line. (From centre drop perpendicular on line, and at point of intersection with circle draw tangent.)

4. Describe a circle of radius 35 millimetres; draw any line in plane of paper; draw a tangent to circle which shall be perpendicular to this line.

5. Draw any line and draw circles of radii 1, 1 and 2 inches, touching the line at any points.

6. Describe two circles of radii 1 inch and 24 inches, so as to touch any line at points 3 inches apart. Do the circles touch one another?

7. A tangent of length 4 inches is drawn from a point to a circle of radius 3 inches. How far is the point from the centre of the circle?

inch, and another

8. A tangent is drawn to a circle of radius circle, concentric with the former, is described of radius 2 inches. What is the length of the tangent between the point where it is intercepted by the second circle and the point of contact? What angle does the intercepted portion of the tangent subtend at the common centre?

9. A circle has a radius of 30 millimetres, and a tangent of length 40 millimetres is drawn to it. What line (curved) represents all the points, outside the circle, from which this tangent may be drawn?

10. From four points, equidistant from one another, on a circle of radius 2 inches, draw tangents to a concentric circle of radius 1 inch.

11. Describe two circles of radii 1 and 1 inches, to touch one another; and describe a circle of radius 23 inches to touch both, and contain both.

12. The preceding problem with each circle external to the other two.

13. Describe three circles of radii 2‡, 3 and 34 inches, so that each may touch the other two.

14. Describe two concentric circles of radii 1 and 3 inches, and describe a number of circles touching both of them.

15. Two circles touch internally at A, and ABC is drawn to meet the circles at B and C. What is the position of radii to B and C with respect to each other? Apply test. Give reasons.

16. Two circles touch the circles at B and C. respect to each other?

externally at A, and ABC is drawn to meet What is the position of radii to B and C with Apply test. Give reasons.

17. OA, OB are drawn through the centre of a circle at right angles to each other, and a tangent to the circle meets these lines at A and B. Two other tangents are drawn to the circle from A and B. What

is the position of these latter tangents with respect to each other? Apply test. Give reasons.

18. Draw two tangents to a circle from an external point, and join the points of contact What is the relation between the angles this "chord of contact" makes with the tangents? Apply test. Give

reasons.

19. Two circles touch externally and parallel diameters are drawn. Lines are drawn from opposite ends of these diameters to the point of contact: what position do they occupy with respect to each

other?

20. Two circles touch internally and parallel diameters are drawn. Lines are drawn from corresponding ends of these diameters to the point of contact: what position do they occupy with respect to each other?

21. Describe two circles with radii 14 in. and in., respectively, their centres being 3 in. apart. Concentric with the larger, describe a third circle of radius in. (1-1); and from the centre of the smallest circle draw a tangent to this third circle. parallel to this tangent, and at distance in. from it. last line with respect to the first two circles? Apply tests.

Draw a line

What is this

22. Describe two circles with radii 11⁄2 in. and in., respectively, their centres being 3 in. apart. Concentric with the larger circle, describe a third circle of radius 12 in. (14+); and from the centre of the smallest circle draw a tangent to this third circle. parallel to this tangent, and at distance in. from it. last line with respect to the first two circles? Apply tests.

Draw a line What is this

CHAPTER XII.

Angles in a Circle.

1. The angles ACB, ADB stand on the same arc AB, the one being at the centre and the other at the circumference.

B

B

B

Measure the number of degrees in each, and compare these numbers.

Make the same constructions in the case of two or three other circles, and repeat the measurements and comparison.

What is your conclusion as to the size of the angle at the centre, compared with the size of the angle at the circumference?