175. Corollary 1. If a line bisects a circle, it is a diameter.

K

1

B

Proof. Let a straight line CA make arc AKC = =arc ARC. Now suppose the center B is not on AC and draw BC and BA. Then since arc CKA = arc CRA, by § 174 one angle A CBA the other angle CBA; that is, Z1=22. Therefore CBA is a straight angle, and hence the line CBA is a straight line; that is, the center B must lie on AC. Therefore AC is a diameter.

R

176. Corollary 2. In the same circle or in equal circles, if two ares are unequal, the greater are subtends the greater central angle. HINT. Prove in a manner like that of § 173.

177. Chord. A chord is a straight line whose extremities are on the circle.

A chord is said to subtend an arc. Every chord, except a diameter, subtends two unequal arcs, a

minor

chord

arc

major are

minor arc and a major arc. If neither is specified, the minor arc

I will be the one meant.

QUERY. Is a diameter a chord ?

Theorem 3

178. In the same circle or in equal circles, if two arcs are equal, their chords are equal.

H

B

Given, in the equal circles whose centers are respectively B and G, the chords AC and FH, subtending the equal arcs AC and FH respectively.

To prove that chord AC chord FH.

=

3. If three diameters, AB, FG, and KR, divide a circle into six equal arcs, prove that the six angles at the center contain 60° each.

4. A, K, B, and R are points in order on a circle with center C such that arc AK equals arc BR. Prove that the angle KCR equals the angle ACB.

5. AB is a chord of a circle equal to its radius. Prove that the arc AB is one sixth of the whole circumference.

6. If the arcs AB, BC, CD, and DE of the same circle are equal, then the chords AC, BD, and CE are equal.

7. K, R, and L are three points on a circle such that arcs KR, RL, and LK are equal. The chords KR, RL, and LK are drawn. Prove that the triangle KRL is equiangular.

Theorem 4 (Converse of Theorem 3)

179. In the same circle or in equal circles, if two chords are equal, their subtended arcs are equal.

Given (using the figure of § 178) the equal circles whose centers are respectively B and G, and chord AC equal to chord FH.

QUERY 1. If two circles are not equal, would equal chords subtend ares equal in length? Explain.

QUERY 2. If two circles are not equal, would arcs equal in length have equal chords? Explain.

EXERCISES

8. If chords AB, BC, CD, and DE are equal, then chords AC, BD, and CE are equal.

9. A, K, B, and R are points in order on a circle such that the chord AB equals the chord KR. Prove that the arc AK equals the arc BR.

10. Two diameters of a circle are perpendicular to each other. Prove that the chords joining their extremities form a square.

Theorem 5

180. In the same circle or in equal circles, if two minor arcs are unequal, the greater are has the greater chord.

Given the equal circles whose centers are respectively C and H, in which minor arc AB is greater than minor arc FG, and chords AB and FG.

11. AK is a diameter and KB and KC are chords such that KB lies between AK and KC. Prove that the chord KB is greater than the chord KC.

12. A diameter of a circle is greater than any other chord of the circle.

Theorem 6 (Converse of Theorem 5)

181. In the same circle or in equal circles, if two chords are unequal, the greater chord has the greater minor arc.

H

Given the equal circles whose centers are respectively C and H, with chord AB greater than chord FG.

13. AB is a diameter and BR and BK are chords with BK the greater. Prove chord AK is less than chord AR.

14. R is the center and AB, BC, and CA are chords of a circle. If AB is greater than BC and BC is greater than CA, prove the angle ARC less than 120°.