the angles of the triangle ABC, and if AO produced meet BC in D, and from O, OE be drawn perpendicular to BC; prove that the angle BOD = the angle COE.

45. The bisectors of the external angles of a quadrilateral form a circumscribed quadrilateral the sum of whose opposite angles is equal to two right angles.

46. The locus of a point equidistant from two given intersecting lines is two lines at right angles to each other. Bisect the adj. Zs between the given lines, etc.

47. Any line is drawn cutting two fixed intersecting lines, and the two angles on the same side of it are bisected: find the locus of the point of intersection of the bisecting lines.

48. The leaf of a book is turned down so that the corner always lies on the same line of printing: find the locus of the foot of the perpendicular from the corner to the crease.

49. ABC is a triangle, D is the middle point of BC, E the middle point of AD; let BE produced meet AC in F: prove that AC is trisected in F.

Draw DG to BF meeting AC in G, etc.

50. D, E, F are the middle points of the sides of a triangle, N is the foot of the perpendicular from the angle opposite D to the side which D bisects: prove that EDF = /ENF.

Let D, E, F lie in BC, CA, AB respectively; ED is II to AB, and DF || to AC, etc.

51. If A, B, C denote the angles of a A, prove that }(A+B), 1(B+C), ¿(C+ A) will be the angles of a ▲ formed by any side and the bisectors of the external angles between that side and the other sides produced.

52. If the exterior angles of a ▲ be bisected, the three external as formed on the sides of the original ▲ are mutually equiangular.

53. If a hexagon have its opposite sides equal and parallel, the three straight lines joining the opposite angles are concurrent.

54. Show that a parallelogram is symmetrical with respect to its centre.

BOOK II.

THE CIRCLE.

DEFINITIONS.

179. A circle is a plane figure bounded by a curved line called the circumference, every point of which is equally distant from a point within called the centre.

A radius is a straight line drawn from the centre to the circumference.

A diameter is a straight line drawn through the centre, and terminated both ways by the circumference.

Thus, in the figure, ABCDE is the circumference; the space included within the circumference is the circle; O is the centre; OA, OB, OC, are radii; AOC is a diameter.

From the definition of a circle, it is evident that all its radii are equal; and also that all its diameters are equal, and each double the radius.

C

E

A

180. An arc of a circle is any part of the circumference, as AED.

A semi-circumference is an arc equal to one-half the cir

cumference.

181. A chord is the straight line which joins any two points on the circumference,* as AD.

*The arc is sometimes said to be subtended by its chord.

Chords of a circle are said to be equally distant from the centre, when the perpendiculars drawn to them from the centre are equal. The chord on which the greater perpendicular falls is said to be further from the centre.

NOTE.-Every chord subtends two arcs whose sum is the circumference. A chord which does not pass through the centre, subtends two unequal arcs. Thus, AD subtends both the arc AED and the arc ABCD; of these, the greater is called the major arc, and the less the minor arc. Thus, the major arc is greater, and the minor arc less than the semi-circumference.

The major and minor arcs, into which a circumference is divided by a chord, are said to be conjugate to each other. When an arc and its chord are spoken of, the minor arc is always meant, unless otherwise stated.

182. A secant of a circle is a straight line of indefinite length which cuts the circum

ference in two points, as AB.

A tangent to a circle is any straight line which meets the circumference, but, being pro- Aduced, does not cut it, as CD. Such a line is said to touch the circle at a point, and the point is called the point of contact,

D

B

or point of tangency.

A secant may be considered as a chord produced, and a chord may be con

sidered as the part of the secant that is within the circle.

If a secant, which cuts a circle at the points P and Q, be gradually turned about P as fixed, the point Q will ultimately approach the fixed point P. When the secant PQB reaches this limiting position, it becomes the tangent to the circle at the point P.

183. Two circles are tangent to each other when they are tangent to the same straight line at the same point.

They are said to have internal contact or external contact, according as one circle is entirely within or entirely without the other.

Circles that have the same centre are said to be concentric.

184. A segment of a circle is the figure included between a chord and its arc, as ACB, or AFB.

A

A segment is called a major or minor D segment, according as its arc is a major or minor arc. The chord of the segment is sometimes called the base of the segment.

с

F

B E

A semicircle is the segment included between a diameter and a semi-circumference, as DFE.

185. A straight line is inscribed in a circle when its extremities are on the circumference, as AC.

A

D

B

186. An inscribed angle is one whose vertex is in the circumference, and whose sides are chords of the circle, as ACB. The AOB, whose vertex is at the centre O, is called the angle at the centre.

187. An angle in a segment is the angle formed by two straight lines drawn from any point in the arc of the segment to the extremities of the base of the segment, as ZACB, or

ADB.

188. A sector of a circle is the figure A bounded by two radii and the arc intercepted between them, as AOB, or BOC.

189. A polygon is said to be inscribed in a circle, when all its vertices are on the circumference, as ABCD.

190. A circle is said to be circumscribed about a polygon, when the circumference passes through each vertex of the polygon.

B

A

B

191. A polygon is said to be circumscribed about a circle, when each of its sides is tangent to the circumference, as ABCD. The circle is then said to be inscribed in the polygon.

B

A

192. From the definition of a circle it follows: (1) The distance of a point from the centre of a circle is less than, equal to, or greater than the radius, according as the point is within, on, or without the circumference.

Hence (157), the locus of a point, which is always at a given distance from a given point, is the circumference of a circle, of which the given point is the centre, and the given distance is the radius.

(2) Circles of equal radii are identically equal. For, if one circle be applied to the other so that their centres coincide, their circumferences will coincide, since all the points of both are at the same distance from the centre (179).

(3) A straight line cannot meet the circumference of a circle in more than two points. For, if it could meet it in three points, these three points would be equally distant from the centre (179). There would then be three equal straight lines drawn from the same point to the same straight line, which is impossible (64).