(2) If the sides of a convex polygon are produced in order, the exterior angles so formed are together equal to four right angles. § 117.

12. THEOREMS ON PARALLELOGRAMS.

(1) The opposite sides and the opposite angles of a parallelogram are equal. § 123.

(2) Either diagonal of a parallelogram divides the figure into two superposable triangles. § 124.

(3) The diagonals of a parallelogram bisect each other.

§ 126.

(4) If two sides of a quadrilateral are parallel and equal, the figure is a parallelogram. § 127.

(5) If both pairs of opposite sides of a quadrilateral are equal, the figure is a parallelogram. § 128.

(6) If two parallelograms have two adjacent sides and the included angle of one equal, respectively, to two adjacent sides and the included angle of the other, the parallelograms are identically equal. § 129.

13. MISCELLANEOUS THeorems.

(1) If two straight lines intersect, the vertical angles are equal. § 58. (2) Not more than two equal line-segments can be drawn from a given point to a given straight line. § 67.

(3) If any pair of alternate angles formed by two straight lines and a transversal are unequal, or if any pair of corresponding angles are unequal, or if the interior angles on one side are not supplementary, the two lines are not parallel, and therefore will meet if produced. § 96.

(4) If the boundaries of one angle are respectively parallel or perpendicular to the boundaries of another, these two angles are either equal or supplementary. § 105.

(5) If the mid-points of the sides of any quadrilateral be joined in order, the figure so formed is a parallelogram, and the sum of the sides of this parallelogram equals the sum of the diagonals of the quadrilateral. § 131.

14. ON SYMMETRY.

(1) If two figures are symmetrical with respect to a straight line, they are superposable by inversion. § 145.

(2) A circle is symmetrical with respect to any of its diameters. § 146.

H

CHAPTER II

THE CIRCLE

SECTION I

DEFINITIONS AND PRELIMINARY THEOREMS

147. In the introductory chapter the following definitions were given :

A circle is a closed line all points of which are equally distant from a certain point within it called the centre.

The line-segment joining the centre to any point of the circle is called a radius, and a line-segment through the centre terminated both ways by the circle is called a diameter.

From these definitions it follows that all radii of the same circle are equal, and all diameters of the same circle are equal.

The distance from the centre of any point inside the circle is less than a radius, and of any point outside the circle is greater than a radius.

148. THEOREM I. Two circles in a plane which have the same centre and equal radii coincide throughout.

For, if there is any point of one circle which does not coincide with a point of the other, it must lie either inside or outside of the other, and hence its distance from the centre is either less or greater than a radius of the other. But this is not the case, since all radii of the same circle are equal by definition and radii of the two circles are assumed to be equal.

COROLLARY I. Two circles in a plane which have the same centre and one point in common coincide throughout.

For they have the same centre and equal radii, since the line joining their common point to the centre is a radius of each circle.

COROLLARY II. Two circles which have equal radii can be made to coincide, and hence are identically equal; and conversely, equal circles have equal radii.

149. THEOREM II. Through a given point any number of different circles can be described.

For, if A is the given point, we may choose any point O whatever for centre, and with radius OA describe a circle which passes through A.

Since a circle is a closed curve, two circles which intersect at one point must also intersect at a second point.

(Art. 25.)

150. THEOREM III. Through two given points any number of different circles can be described.

Let A and B be the given points; then every point of the perpendicular bisector of the line-segment AB is equidistant from A and B (Art. 72). Choose any point O on this perpendicular bisector for centre, and the circle described with radius OA will pass through both A and B.

151. THEOREM IV. Through three given points not in the same straight line, one and only one circle can be described.

Let A, B, and C be the three given points. Then the locus of points equidistant from A and B is the perpendicular bisector of the line-segment AB, and the locus of points equidistant from B and C is the perpendicular bisector of the line-segment BC. These two loci have one (Art. 98) and only one common point; which point, O say, is equidistant from A, B, and C, and is the only such point.

With O as centre and with radius OA a circle may be described through A, B, and C, while no other circle can be

described through these three points, since no other point than O can be found for centre.

COROLLARY I. Two circles which coincide at three points coincide throughout.

For they must have the same centre and equal radii.

COROLLARY II. Two different circles can have at most two points in common;

Or, two circles can intersect in at most two points.

Incidentally in this article we have solved the problem:

To find the centre of the circle which passes through three given points, or of which at least three points are given. This is the same problem as To pass a circle through the three vertices of a triangle, a thing which can always be done.

152. The questions arise:

Can a circle be described through three given points which lie in a straight line?

If you should proceed as in the case where the three points do
not lie in one straight line, in what particular would the con-
struction fail?

How many points of a straight line are equidistant from any
given point?
(See Art. 67.)

THEOREM V. A straight line can intersect a circle in at most two points;

Or, what is the same thing, a straight line can have at most two points in common with any circle.

153. DEFINITIONS. Any portion of a circle terminated by two points is called an arc of the circle.

The straight line joining any two points of a circle, i.e. joining the extremities of an arc, is called a chord of the circle.

The chord is said to subtend the arc, the arc to be subtended by the chord..

Every chord subtends two arcs of the circle, one on either side of it, called the greater or major arc and the lesser or minor

arc.

The two arcs subtended by any chord together make up the whole circle. Each of these arcs is called the conjugate of the other.

When we speak of the arc subtended by a given chord we shall always have in mind the lesser or minor arc unless the contrary is expressly stated.

154. The arc of a circle subtended by a diameter is called a semicircle.

Since a circle is symmetrical with respect to any of its diameters (Art. 146), a semicircle and its conjugate semicircle are equal. If you should fold the circle over, along the diameter, the two semicircles would coincide.

Hence any semicircle is half of a circle, and all semicircles belonging to the same or equal circles are equal.

155. THEOREM VI. The perpendicular bisector of any chord of a circle passes through the centre.

For it is the locus of points equidistant from the extremities of the chord, and the centre being equidistant from the extremities of the chord must lie on this locus.

156. DEFINITIONS.

The figure formed by an arc of a

circle and its subtending chord is called a segment of the circle.

The figure formed by an arc and the two radii to its extremities is called a sector of the circle. The angle at the centre formed by two radii is called the angle at the centre subtended by the intercepted arc, or by the chord of that arc.

Segment

Sector