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BOOK II.

CIRCLES.

DEFINITIONS. 160. DEF. A Circle is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the Centre.

161. DEF. The Circumference of a circle is the line which bounds the circle.

162. DEF. A Radius of a circle is any straight line drawn from the centre to the circumference, as 0 A, Fig. 1.

163. DEF. A Diameter of a circle is any straight line passing through the centre and having its extremities in the circumference, as A B, Fig. 2.

By the definition of a circle, all its radii are equal. Hence, all its diameters are equal, since the diameter is equal to twice the radius.

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P.

D
Fig. 1.
Fig. 2.

Fig. 3. 164.. DEF. An Arc of a circle is any portion of the circumference, as A M B, Fig. 3.

165. DEF. A Semi-circumference is an arc equal to one half the circumference, as A M B, Fig. 2.

166. DEF. A Chord of a circle is any straight line having its extremities in the circumference, as A B, Fig. 3.

Every chord subtends two arcs whose sum is the circumference. Thus the chord A B, (Fig. 3), subtends the arc A M B and the arc A D B. Whenever a chord and its arc are spoken of, the less arc is meant unless it be otherwise stated.

167. DEF. A Segment of a circle is a portion of a circle enclosed by an arc and its chord, as A M B, Fig. 1.

168. DEF. A Semicircle is a segment equal to one half the circle, as A DC, Fig. 1.

169. DEF. A Sector of a circle is a portion of the circle enclosed by two radii and the arc which they intercept, as A C B, Fig. 2.

170. DEF. A Tangent is a straight line which touches the circumference but does not intersect it, however far produced. The point in which the tangent touches the circumference is called the Point of Contact, or Point of Tangency.

171. DEF. Two Circumferences are tangent to each other when they are tangent to a straight line at the same point.

172. DEF. A Secant is a straight line which intersects the circumference in two points, as A D, Fig. 3.

D
Fig. 1.

Fig. 2

Fig. 3.

Fig. 4.

173. DEF. A straight line is Inscribed in a circle when its extremities lie in the circumference of the circle, as A B, Fig. 1.

An angle is inscribed in a circle when its vertex is in the circumference and its sides are chords of that circumference, as Z A B C, Fig. 1.

A polygon is inscribed in a circle when its sides are chords of the circle, as A A BC, Fig. 1.

A circle is inscribed in a polygon when the circumference touches the sides of the polygon but does not intersect them, as in Fig. 4.

174. DEF. A polygon is Circumscribed about a circle when all the sides of the polygon are tangents to the circle, as in Fig. 4.

A circle is circumscribed about a polygon when the circumference passes through all the vertices of the polygon, as in Fig. 1.

175. DEF. Equal circles are circles which have equal radii. 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.

176. Every diameter bisects the circle and its circumference. For if we fold over the segment A M B on A B as an axis until it comes into the plane of A PB, the arc A M B will coincide with the arc A PB; because every point in each is equally distant from the centre 0.

PROPOSITION I. THEOREM. 177. The diameter of a circle is greater than any other chord.

Let A B be the diameter of the circle

A MB, and AE any other chord.
We are to prove A B > A E.

At
From C, the centre of the O, draw C E.

CE=CB,
(being radii of the same circle).
But
AC + CE > A E,

, § 96
(the sum of two sides of a A > the third side).
Substitute for C E, in the above inequality, its equal C B.
Then

AC + CB > A E, or
A B > A E.

Q. E. D.

PROPOSITION II. THEOREM. 178. A straight line cannot intersect the circumference of a circle in more than two points.

M

Let HK be any line cutting the circumference A MP.

We are to prove that H K can intersect the circumference in only two points.

If it be possible, let H K intersect the circumference in three points, H, P, and K.

Then

§ 163

From 0, the centre of the O, draw the radii 0 H, OP, and O K.

OH, O P, and 0 K are equal,

(being radii of the same circle). .. if H K could intersect the circumference in three points, we should have three equal straight lines 0 H, OP, and 0 K drawn from the same point to a given straight line, which is impossible,

§ 56 (only two equal straight lines can be drawn from a point to a straight line).

. a straight line can intersect the circumference in only two points.

Q. E. D.

PROPOSITION III. THEOREM. 179. In the same circle, or equal circles, equal angles at the centre intercept equal arcs on the circumference.

PI

In the equal circles ABP and A'B'P' let 20=20'.

We are to prove arc R S = arc R' S'.

Apply O ABP to 0 A' B' P',
so that 20 shall coincide with 2 0'.

The point R will fall upon R',

§ 176 (for O R= 0' R', being radii of equal ©),

and the point S will fall upon s', § 176

(for OS = 01 S', being radii of equal ©). Then the arc R S must coincide with the arc RS'. For, otherwise, there would be some points in the circumference unequally distant from the centre, which is contrary to the definition of a circle.

$ 160

Q. E. D.

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