2. A line of fixed length remains always parallel to itself while one extremity describes a circle, what is the locus of the other extremity?

3. If a straight line cut two concentric circles, the parts intercepted between the circumferences are equal.

4. Through any point within a circumference chords are drawn equally inclined to the diameter through that point, shew that they are equal.

5. The same if the point be without the circumference.

6. Find the longest and shortest lines that can be drawn between two given circumferences.

7. If two equal circumferences intersect at right angles, the common chord is equal to the distance of the centres. Circumferences are said to intersect at right angles when the tangents at the point of section are at right angles.

8. Under what conditions will one circumference cut, touch, or enclose another?

9. A circle being given, how many circles of the same radius will enclose it?

10. Through a given internal point to draw the shortest possible chord.

11. If two circumferences intersect and parallel chords be drawn through the points of section, these chords are equal.

12. If two chords intersect within a circle, the angle which is contained between them is equal to half the sum of the angles subtended at the centre by the arcs which they intercept.

13. If the two chords intersect without the circle, to what is the angle contained between them equal?

14. ABC is a triangle, and B is greater than C, on AC take AD = AB and join BD. Shew that ADB is equal to half the sum of B and C, and DBC to half their difference.

15. The lines joining the extremities of two diameters are parallel.

16. If triangles be formed by joining the extremities of intersecting chords, these triangles are equiangular to one another.

The chords may intersect either within or without the circle, and their extremities may be joined in different ways.

17. If parallel straight lines meet a circumference, they cut off equal arcs.

18. What is the locus of a point from which a given line is always seen under a constant angle?

19. ABC is a triangle, AD, BE, CF perpendiculars let fall on the opposite sides. Shew that these bisect the angles of the triangle DEF.

20. If two circles touch each other and secants be drawn through the point of contact, the lines joining their extremities are parallel.

21. AB is the diameter of a circle, AC, any chord, is produced to M so that CM is equal to CB. What is the locus of M?

22. Chords are drawn through a fixed point, find the locus of the middle points of these.

23. A line is drawn from a fixed point to a circumference, find the locus of its middle point.

24. Shew that a circle may be described about any regular polygon; i. e. any polygon having equal sides and angles.

25. Can a circle be described about any quadrilateral figure?

CONSTRUCTIONS.

1. Three points being given, find the centre of the circle which passes through them.

When does the construction fail?

[I. 19. 21.

2. The circumference of a circle being given, find its

centre.

3. Construct a circle, passing through two points, and having

1. Its centre on a given straight line.

2. Its centre on a given circumference.

3. Its radius equal to a given straight line.

4. From a given point to draw tangents to a given circle.

The point being without the circumference, draw a line joining it with the centre; on this line as diameter describe a circle, which will cut the given circumference in two points. The lines joining these with the given point are the tangents required. Prove, that these lines are tangents, that they are equal, that they subtend equal angles at the centre, and that they are equally inclined to the diameter passing through the external point.

If the given point be on the circumference?

5. Construct three circumferences of equal radius so as to touch one another, and draw another circumference touching all three. Two cases.

6. Bisect a given arc.

7. From a given circle to cut off a segment containing an angle equal to a given angle.

[II. 16.

8. On a given line to construct a segment of a circle containing an angle equal to a given angle.

[II. 16.

9. From a given point to draw a secant to a given circumference, so that the chord intercepted may have a given length.

10. To construct a right-angled triangle, knowing,

[II. 12.

1. The hypothenuse and one of the acute angles.

2. The hypothenuse and a side.

11. To construct an isosceles triangle knowing the base, and the angle at the vertex.

12. To construct a square knowing the diagonal.

13. To construct a circle of given radius,

[II. 14.

1. Passing through a given point, and touching a

given line, or a given circumference.

2. Touching two lines.

3. Touching a straight line and a circumference.

4. Touching two circumferences.

14. To draw a common tangent to two given circumferences. How many such tangents can be drawn?

A circle is said to be inscribed in a rectilineal figure when the circumference touches every side of the figure, and to be described about such a figure when the circumference passes through all the angular points of the figure.

15. To describe a circle about a given triangle.

16. To describe a circle about a given square.

17. To inscribe a circle in a given triangle.

[1. 20. 22.

18. To construct a circle touching three given straight

lines.

The three straight lines are supposed to meet so as to form a triangle, and there are four circles satisfying the conditions. Shew that the lines which join the centres of the exterior circles pass through the angular points of the triangle; also that the tangents drawn from the angular points to the further exterior circles are all equal.

19. In a given circle inscribe,

1. An equilateral triangle.

2. A square.

3. A regular hexagon.

[II. 16.

[II. 15. COR. 2.

20. Construct a circle touching a given line in a given

point, and

1. Passing through a given point.

2. Having a given radius.

3. Touching another given line.