PLANE GEOMETRY

BOOK II

THE CIRCLE

150.* Def.—A circle is a plane figure bounded by a line, all points of which are equally distant from a point within called the centre.

151.* Defs.-The line which bounds the circle is called

its circumference.

An arc is any part of a circumference.

152.* Def.—Any straight line from the centre to the circumference is a radius.

By the definition of a circle all its radii are equal.

153. Def.-A chord is a straight line having its extremities in the circumference.

154. Def.-A diameter is a chord through the centre. All diameters are equal, each being twice a radius.

155. Defs.-A sector is that portion of a circle bounded by two radii and the intercepted arc.

SECTOR

The angle between the radii is called the angle of the

sector.

*These definitions are repeated from § 20.

156. Def.-Concentric circles are circles which have the same centre.

PROPOSITION I. THEOREM

15%. The diameter of a circle is greater than any other chord.

GIVEN the circle ABC and AC, any chord not a diameter.

158. Circles which have equal radii are equal, and if their centres be made to coincide they will coincide throughout; conversely, equal circles have equal radii.

c'

I. GIVEN any two circles, C and C' with centres O and Ở and equal

radii.

TO PROVE

the circles C and C' are equal.

Place the circles so that O falls on O'.

Then the circumference of C will coincide with the circumference of C'.

For, if any portion of one fell without the other, its distance from the centre would be greater than the distance of the other. Hence the radii would be unequal, which is contrary to the hypothesis.

Ax. 10

Therefore, the circumferences coincide, and the circles coincide and are equal.

Q. E. D.

Since the circles are equal they can be made to coincide, and therefore their radii will coincide, and are equal. Q. E. D.

159. COR. I. Hence, if a circle be turned about its centre as a pivot, its circumference will continue to occupy the same position.

160. COR. II. The diameter of a circle bisects the circle and the circumference.

Hint.-Fold over on the diameter as an axis.

161. Defs.-The halves into which a diameter divides. a circle are called semicircles, and the halves into which it divides the circumference are called semicircumfer

ences.

PROPOSITION III. THEOREM

162. In the same circle or equal circles, equal angles at the centre intercept equal arcs; conversely, equal arcs are intercepted by equal angles at the centre.

TO PROVE

I. GIVEN equal circles and equal angles at their centres, O and Ơ.

arc AB arc A'B'.

Apply the circles making the angle O coincide with angle O'.

A will coincide with A', and B with B'.

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[For A0=A'0', and OB=O'B', being radii of equal circles.] Then the arc AB will coincide with the arc A'B', and is equal to it.

II. CONVERSELY:

GIVEN equal circles having equal arcs AB and A'B'.

TO PROVE the subtended angles O and O' equal.

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Q. E. D.

Apply the circles making the arc AB coincide with its equal A'B'.

Then AO will coincide with A'O', and BO with B'O' Ax. a Therefore angles O and O' coincide and are equal.

Q. E. D.

163. Exercise.-In the same circle or equal circles equal angles at the centre include equal sectors, and conversely.

The proof is analogous to the preceding, requiring “sector" in place of "arc."

FROPOSITION IV. THEOREM

164. In the same circle or equal circles, equal chords subtend equal arcs; conversely, equal arcs are subtended by equal chords.

GIVEN equal circles, O and O', and equal chords, AB and A'B'.

Draw the four radii OA, OB, O'A', O'B'.

In the triangles AOB and A'O'B'