PROPOSITION VI

170. In the same circle, or in equal circles, equal chords are equidistant from the centre; and of two unequal chords, the greater is nearer to the centre than the less.

1

First, let O, and O2 be the centres of two equal circles, AВ1 and AВ equal chords in them.

It is required to prove that the perpendiculars OC1 and OC2 from the centre upon the chords are equal.

2

Proof. The angles at the centre A,OВ ̧ and АОВ2 are equal. Give reasons. Also A111 = ZA2O2C2. Why? AAOC1 is identically equal to ▲ A202C2. Prove. Therefore

Next, let the chord AD2 be greater than the chord AB.

2

It is required to prove that the perpendicular O2C2 upon the lesser chord is greater than the perpendicular OE, upon the greater chord.

Proof. Since the chord A,D, is greater than the chord AB, the arc A,D, is greater than the arc A,B2.

(Prop. IV.)

If the chords be so placed as to coincide at one extremity and lie on the same side of the centre, the arc A,D, will extend beyond the arc A,B, and the chord AB, will lie on the opposite side of A,D, from the centre. Consequently, the perpendicular OC2 will intersect the chord AD, at some point F2.

2

Now OC is greater than O,F2, and OF2 is greater than O¿E2. Why? Therefore O2C2 is greater than O2E2.

PROPOSITION VII

171. In the same circle, or in equal circles, chords equidistant from the centre are equal; and of two chords unequally distant, the one nearer the centre is the greater.

What is the relation of this proposition to Proposition VI?

The proof is left to the pupil, with the suggestion that the indirect method will probably be easiest.

Thus, if in the first case, the chords are not equal, one or the other of them must be nearer the centre. Which? Prop. VI.

In the second case, if the one nearer the centre is not the greater, what?

172. COROLLARY.

be drawn in any circle.

A diameter is the greatest chord that can

This follows directly from the above proposition, or it may be shown otherwise, as follows: :

The diameter AB equals the sum of the radii AO and OC, which is greater than any chord AC, not a diameter.

A

EXERCISES

1. If an equilateral triangle is inscribed in a circle, the sides are equidistant from the centre.

2. If two chords of a circle which intersect make equal angles with the line joining their common point to the centre, show that the chords are equal.

3. A chord which is perpendicular to a radius is less than any other chord through their point of intersection.

4. If from a point within a circle more than two equal straight lines can be drawn to the circle, that point must be the centre.

SUGGESTION.

Suppose that three equal lines can be drawn from the point, and show that the point is the intersection of two lines upon each of which the centre lies. See Art. 155.

5. Describe two concentric circles each of which passes through two given points, the first through A and B, say, and the second through C and D.

PROPOSITION VIII

173. Of all line-segments which can be drawn to a circle from a point within it, not the centre, the greatest is that which passes through the centre, and the least is that which, if produced backward, would pass through the centre; and of any two others, the greater is that which makes the less angle with the greater segment of the diameter through the point.

Let S be any point within a circle other than the centre, SA the line drawn from S through the centre to the circle, SB the line from S to the circle which, if produced backward, would pass through the centre, SC and SD any other straight lines from S to the circle, of which SC makes a less angle with the diameter through S than does SD.

It is required to prove (1) that SA is the greatest line from S to the circle, (2) that SB is the least line, and (3) that SC is greater than SD.

Proof. Join OC and OD.

First, SA equals the sum of SO and OC.

But the sum of SO and OC is greater than SC. Why? Therefore SA is greater than SC, any other line drawn from S to the circle.

Next, SB equals the difference between SO and OC.

But the difference between SO and OC is less than SC.

(Ex. 3, p. 48.)

Therefore SB is less than SC, any other line drawn from S to the circle.

Lastly, if we rotate the line SC about S into the position SD, thereby increasing ASC (hypothesis), we at the same time increase AOC till it becomes AOD.

Since AOC is less than AOD, the supplementary

SOC

is greater than the supplementary SOD, while the sides containing these angles are respectively equal.

Therefore SC is greater than SD.

(Art. 79.)

174. COROLLARY. From any point within a circle two equal straight lines can be drawn to the circle; these make equal angles with the diameter through the point.

EXERCISES

1. State and prove a theorem for a point without a circle similar to that of Proposition VIII.

2. If two circles intersect, any two parallel lines drawn through the points of intersection and terminated both ways by the circles are equal. 3. If two circles intersect, any two lines drawn through one point of intersection, making equal angles with the line of centres and terminated both ways by the circles, are equal.

4. If with the vertex of an isosceles triangle for centre a circle is described which cuts the base or the base produced, show that the segments of the base line intercepted between the extremities of the base and the circle are equal.

175. In Proposition VIII we saw that if from S, any point within a circle, not the centre, a straight line SP is drawn to the circle, and the line rotated about S in the way indicated by the arrowhead, while P traverses the circle, the magnitude SP will continuously increase till P reaches the point A. After this it will continuously decrease till P comes to coincide with B, when it will again begin to increase.

B

The magnitude SP is thus a variable quantity, varying continuously as the angle ASP varies continuously. It has a maximum value, viz. SA, and a minimum value, viz. SB.

If the point S is chosen on the circle, SP is a variable chord whose maximum value is a diameter and whose minimum value is zero.

SECTION II

ANGLES INSCRIBED IN ARCS

176. DEFINITION. If a point is chosen on any arc of a circle, and the chords are drawn from it to the extremities of the arc, the angle between these chords is said to be an angle in the arc, or an angle inscribed in the arc, and the arc is said to contain the angle.

An angle in an arc is often spoken of as an angle in the segment formed by the arc and its chord, and the segment then is said to contain the angle.

The angle ACB is said to be inscribed in the arc AEB, or in the segment AEB; it is an angle at a point of the circle subtended by the arc AFB, or subtended by the chord AB.

Sometimes the expression 'an angle stands upon an arc' is used instead of 'is subtended by an arc.'

C

E

D

B

F

EXERCISE. In what arc is ABC inscribed? What arc does it subtend? What chord does it subtend?

177. DEFINITION. A polygon is said to be inscribed in a circle when its vertices lie on the circle; and the circle is said to be circumscribed about the polygon.

Each angle of an inscribed polygon is subtended by an arc of the circle, and is inscribed in the conjugate arc.