BOOK II

THE CIRCLE

191. A curved line is a line no part of which is straight.

192. A circumference is a curved line every point of which is equally distant from a point within, called the center.

193. A circle is a portion of a plane bounded by a circumference. [0.]

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

A diameter is a straight line containing the center, and whose extremities are in the circumference.

A secant is a straight line cutting the circumference in two points.

A chord is a straight line whose extremities are in the circumference.

A tangent is a straight line which touches the circumference at only one point, and does not cut it, however far it may be extended. The point at which the line touches the circumference is called the point of contact or the point of tangency.

195. A central angle is an angle formed by two radii.

An inscribed angle is an angle whose vertex is on the circumference and whose sides are chords.

196. An arc is any part of a circumference.

A semicircumference is an arc equal to half a circumference.

A quadrant is an arc equal to one fourth of a circumference. Equal circles are circles having equal radii.

Concentric circles are circles having the same center.

197. A sector is the part of a circle bounded by two radii and their included arc.

A segment of a circle is the part of a circle bounded by an arc and its chord.

A semicircle is a segment bounded by a semicircumference and its diameter.

198. Two circles are tangent to each other if they are tangent to the same line at the same point. Circles may be tangent to each other internally, if the one is within the other, or externally, if each is without the other.

199. POSTULATE. A circumference can be described about any given point as center and with any given line as radius.

Explanatory. A circle is named either by its center or by three points on its circumference, as "the Oo," or "the O ABC."

The verb to subtend is used in the sense of "to cut off."

A chord subtends an arc. Hence an arc is subtended by a chord.

An angle is said to intercept the arc between its sides. Hence an arc is intercepted by an angle.

The hypothesis is contained in what constitutes the subject of the principal verb of the theorem. (See 59.)

PRELIMINARY THEOREMS

200. THEOREM. All radii of the same circle are equal. (See 192.) 201. THEOREM. All radii of equal circles are equal. (See 196.) 202. THEOREM. The diameter of a circle equals twice the radius. 203. THEOREM. All diameters of the same or equal circles are equal. (Ax. 3.)

204. THEOREM. The diameter of a circle bisects the circle and the circumference.

Given: Any O and a diameter.

To Prove The segments formed are equal, that is, the diameter bisects the circle and the circumference.

:

Proof Suppose one segment folded over upon the other segment, using the diameter as an axis. If the arcs do not coincide, there are points of the circumference unequally distant from the center. But this is impossible (?) (192). .. the segments coincide and are equal (?) (28). Q.E.D. 205. THEOREM. With a given point as center and a given line as radius, it is possible to describe only one circumference. (See 192.) That is, a circumference is determined if its center and radius are fixed.

NOTE. The word "circle" is frequently used in the sense of “circumference." Thus one may properly speak of drawing a circle. The established definitions could not admit of such an interpretation save as custom makes it permissible.

Ex. Draw two intersecting circles and their common chord. Draw two circles which have no common chord. Draw figures to illustrate all the nouns defined on the two preceding pages.

THEOREMS AND DEMONSTRATIONS

206. THEOREM. In the same circle (or in equal circles) equal central angles intercept equal arcs.

ӨӨ

Proof: Superpose Oo upon the equal c, making o coincide with its equal, c. Point 4 will fall on L, and point B on M (?) (201).

Arc AB will coincide with arc LM (?) (192).

.. AB=LM (?) (28).

Q.E.D.

207. THEOREM. In the same circle (or in equal circles) equal arcs are intercepted by equal central angles. [Converse.]

Given: 0 =O C; arc AB arc LM.

To Prove: 40=ZC.

Proof: Superpose O upon the equal C C, making the centers coincide and point A fall on point L. Then arc AB will coincide with arc LM and point B will fall on point M. (Because the arcs are

=

:.)

Hence 04 will coincide with CL, and OB with CM (?) (39). ..20=2C (?) (28).

Q.E.D.

Ex. 1. Can arcs of unequal circles be made to coincide? Explain.
Ex. 2. If two sectors are equal, name the several parts that must be

equal.

208. THEOREM. In the same circle (or in equal circles):

I. If two central angles are unequal, the greater angle intercepts the greater arc.

II. If two arcs are unequal, the greater arc is intercepted by the greater central angle. [Converse.]

M

I. Given: 0 = 0 C; ≤ LCM > ≤ 0.

To Prove: Arc LM > are AB.

Proof: Superpose ○ o upon O C, making sector AOB fall in position of sector XCM, OB coinciding with CM.

CX is within the angle LCM (≤ LCM > ≤ 0).

Arc AB will fall upon LM, in the position XM (192).

.'. arc LM > arc XM (Ax. 5).

II. Given: (?). To Prove:

That is, arc LM > arc AB.

LCM > 0.

Q.E.D.

Proof: The pupil may employ either superposition, as in I, or the method of exclusion, as in 87.

NOTE. Unless otherwise specified, the arc of a chord always refers to the lesser of the two arcs. If two arcs (in the same or equal circles) are concerned, it is understood either that each is less than a semicircumference, or each is greater.

Ex. 1. Two sectors are equal if the radii and central angle of one are equal respectively to the radii and central angle of the other.

Ex. 2. If in the figure of 206, arcs AB and LM were removed, how would the remaining arcs compare?

Ex. 3. If in the figure of 208, arcs AB and LM were removed, how would the remaining arcs compare?