BOOK II

THE CIRCLE

159. Circle. A closed curve lying in a plane, and such that all of its points are equally distant from a fixed point in the plane, is called a circle.

160. Circle as a Locus. It follows that the locus of a point in a plane at a given distance from a fixed point is a circle.

161. Radius. A straight line from the center to the circle is called a radius.

162. Equal Radii. It follows that all radii of the same circle or of equal circles are equal, and that all circles of equal radii are equal.

163. Diameter. A straight line through the center, terminated at each end by the circle, is called a diameter.

Since a diameter equals two radii, it follows that all diameters of the same circle or of equal circles are equal.

164. Arc. Any portion of a circle is called an arc.

An arc that is half of a circle is called a semicircle.

An arc less than a semicircle is called a minor arc, and an arc greater than a semicircle is called a major arc. The word arc taken alone is generally understood to mean a minor arc.

165. Central Angle. If the vertex of an angle is at the center of a circle and the sides are radii of the circle, the angle is called a central angle.

An angle is said to intercept any arc cut off by its sides, and the arc is said to subtend the angle.

PROPOSITION I. THEOREM

166. In the same circle or in equal circles equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc.

Given two equal circles with centers O and O', with angles AOB and A'O'B' equal, and with angle AOC greater than angle A'O'B'. To prove that arc AB = arc A'B';

1.

2. arc AC > arc A'B'.

Proof. 1. Place the circle with center O on the circle with center O' so that ZAOB shall coincide with its equal, ▲ A'O'B'. In the case of the same circle, swing one angle about O until it coincides with its equal angle.

Post. 5

Then A falls on A', and B on B'.
(Radii of equal circles are equal.)

§ 162

... arc AB coincides with arc A'B'.

§ 159

(Every point of each is equally distant from the center.)

PROPOSITION II. THEOREM

167. In the same circle or in equal circles equal arcs subtend equal central angles; and of two unequal arcs the greater subtends the greater central angle.

Given two equal circles with centers O and O', with aros AB and A'B' equal, and with arc AC greater than arc A'B'.

To prove that 1. ZAOB=LA'O'B';

2. ZAOC> <A'O'B'.

Proof. 1. Using the figure of Prop. I, place the circle with center on the circle with center O' so that OA shall fall on its equal 'A', and the arc AB on its equal A'B'.

Then OB coincides with O'B'.

.. LAOB=LA'O'B'.

Post. 5

Post. 1

§ 23

Proof. 2. Since arc AC>arc A'B', it is greater than arc AB, the equal of arc A'B', and OB lies within the

if

.. LAOC>LAOB.

AOC.

Ax. 9

Ax. 11

Q. E.D.

.. ZAOC > ZA'O'B', by Ax. 9.

This proposition is the converse of Prop. I.

168. Law of Converse Theorems. Of four magnitudes, a, b, x, y,

(1) a>b when x>y,,

(2) α= b when x = y,

and

(3) a <b when x <y,

then the converses of these three statements are always true.

For when a > b it is impossible that x=y, for then a would equal b by (2); or that x <y, for then a would be less than b by (3). Hence x > y when a > b. In the same way, xy when a = b, and x <y when a <b. 169. Chord. A straight line that has its extremities on a circle is called a chord.

A chord is said to subtend the arcs that it cuts from a circle. Unless the contrary is stated, the chord is taken as subtending the minor arc.

ARC

MINOR

CHORD

MAJOR

170. In the same circle or in equal circles, if two arcs are equal, they are subtended by equal chords; and if two arcs are unequal, the greater is subtended by the greater chord.

B

Α'

Given two equal circles with centers O and O', with arcs AB and A'B' equal, and with arc AF greater than arc A'B'.

To prove that 1. chord AB chord A'B';

=

2. chord AF> chord A'B'.

Proof. 1. Draw the radii OA, OB, OF, O'A', O'B'.

Since

and

and

but

Proof. 2. In the A OAF and O'A'B',

OAO'A', and OF O'B',

ZAOF is greater than ▲ A'O'B'.

§ 162

§ 167

§ 68

§ 67

$ 162

§ 167

(In equal, of two unequal arcs the greater subtends the greater central ≤.)

.. chord AF chord A'B', by § 115.

Q. E.D.

171. COROLLARY. In the same circle or in equal circles, the greater of two unequal major arcs is subtended by the less chord.

PROPOSITION IV. THEOREM

172. In the same circle or in equal circles, if two chords are equal, they subtend equal arcs; and if two chords are unequal, the greater subtends the greater arc.

Α'

Given two equal circles with centers O and O', with chords AB and A'B' equal, and with chord AF greater than chord A'B'.

Since

Proof. 1. Draw the radii OA, OB, OF, O'A', O'B'.

and

OA = O'A', and OB=0'B',

chord AB = chord A'B',

162

Given

..A OAB is congruent to AO'A'B',

§ 80

Q. E. D.

.'. arc AF >arc A'B', by § 166.

This proposition is the converse of Prop. III.

173. COROLLARY. In the same circle or in equal circles the greater of two unequal chords subtends the less major arc.